Groups, Designs and Linear Algebra: Orbit incidence matrices 0 G = h(0, 3, 2)(1, 4, 6), (1, 4)(2, 3)i = (0)(1)(2)(3)(4)(5)(6) (1, 4)(2, 3) (0, 2, 3)(1, 6, 4) (0, 2)(4, 6) (0, 3, 2)(1, 4, 6) (0, 3)(1, 6) 046 016 134 014 136 346 246 146 126 124 1 4 5 2 6 3 036 045 026 356 125 123 126 015 056 035 013 034 145 345 135 235 024 236 234 012 023 456 256 245 025 {46, 16, 14} 36, 34, 26 12, 04, 01 {24, 13, 06} 1 1 2 0 0 0 1 0 0 0 0 1 1 1 1 0 0 1 0 0 0 0 2 2 0 0 0 0 1 0 {23, 03, 02} 0 0 0 2 1 1 0 0 0 1 {56, 45, 15} 0 0 0 0 0 0 2 2 1 0 {35, 25, 05} 0 0 0 0 0 0 0 2 1 2 CIMPA School: Modern Methods in Combinatorics ECOS 2013 1 Donald L. Kreher MICHIGAN TECHNOLOGICAL UNIVERSITY San Luis Argentina 2 Contents Part 1. t-wise balanced designs 1. Introduction 2. Formalities 2.1. Designs 2.2. Orbits 2.3. Orbit incidence matrix 3. Available ingredients Research Problem 1 Research Problem 2 4. References 3 3 4 4 4 5 7 9 9 9 Part 2. Graphical Designs 5. Introduction 6. Orbits of Sn acting on E(Kn ). 7. Steiner graphical designs. 8. Bigraphical designs 8.1. Remarks on the 5–(16, {6, 8}, 1) design. 9. Multi-graphical designs Research Problem 3 Research Problem 4 10. References on graphical designs 10 10 10 11 18 18 19 19 19 19 Part 11. 12. 13. 14. 15. 16. 17. 21 21 21 22 23 24 25 26 28 28 Part 19. 20. 21. 29 29 30 36 42 42 43 3. Incidence Algebra Matrices and Relations Group actions Algebra of G-invariant matrices. Special matrices Fusion. The fundamental epimorphism. Some applications Research Problem 5 18. References 4. The hole size bound Review and introduction Incomplete t-wise balanced designs. Bounds are sharp Research Problem 6 22. References Index 3 Part 1. t-wise balanced designs 1. Introduction Given a set X = {x1 , x2 , . . . , xv } of v-points and a positive integer t < v we ask: Can a collection B = {B1 , B2 , . . . , Bb } of subsets of X be found so that every t-element subset of X is contained in exactly one of them? Gino Fano 1871–1952 A popular example called the Fano plane (after Gino Fano) is provided in Example 1.1. {0, 1, 2}, {0, 3, 4}, {0, 5, 6}, {1, 3, 5}, {1, 4, 6}, Example 1.1. For t = 2, X = {0, 1, 2, 3, 4, 5, 6}, and B = . {2, 3, 6}, {2, 4, 5} Every pair is in exactly one of these chosen subsets. 012 034 056 135 146 236 245 √ 01 √ ··· ··· ··· ··· ··· ··· 0 02 ·√· · ··· ··· ··· ··· ··· 03 · · · · · · · · · · · · · · · · · · √ 04 · · · ·√· · ··· ··· ··· ··· 1 4 05 · · · ··· ··· ··· ··· ··· √ 06 ·√· · ··· ··· ··· ··· ··· 5 12 ··· ··· ·√· · ··· ··· ··· 2 3 6 13 · · · ··· ··· ·√· · ··· ··· 14 · · · ··· ··· ·√· · ··· ··· G = h(0, 3, 2)(1, 4, 6), (1, 4)(2, 3)i 15 · · · ··· ··· ·√· · ··· ··· (0)(1)(2)(3)(4)(5)(6) 16 · · · ··· ··· ··· ·√· · ··· (1, 4)(2, 3) 23 · · · ··· ··· ··· ··· ·√· · (0, 2, 3)(1, 6, 4) 24 · · · ··· ··· ··· ··· ··· √ = (0, 2)(4, 6) 25 · · · ··· ··· ··· ··· ·√· · (0, 3, 2)(1, 4, 6) ·√· · ··· ··· ··· ··· 26 · · · (0, 3)(1, 6) 34 · · · ··· ·√· · ··· ··· ··· 35 · · · ··· ··· ··· ·√· · ··· is an obvious automorphism group. 36 · · · ··· ··· ··· ··· ·√· · 45 · · · ··· ··· ··· ·√· · ··· 46 · · · ··· ·√· · ··· ··· ··· 56 · · · ··· ··· ··· ··· ··· In Example 1.1 we list chosen subsets Bj along the top of an array whose rows are labeled by pairs of points Pi . If Pi ⊆ Bj a check (or tick) mark is drawn. We see from this array that indeed every pair is contained in exactly one of the chosen subsets. Adjacent to the array a figure is drawn representing this system of subsets. The six lines and one circle represent the 7 seven chosen subsets. From this figure it is clear that the group G that we provide is an automorphism group. We now ask: Given a permutation subgroup G of Sym(X), the symmetric group of all permutations on X, does there there exist a collection B = {B1 , B2 , . . . , Bb } 4 of subsets of X so that every t-element subset of X is contained in exactly one of them, where G is an automorphism group of B. Before we can attempt to answer these questions we first provide some formal notation and definitions. 2. Formalities 2.1. Designs. A t-wise balanced design (tBD) with parameters t-(v, K, λ) is a pair (X, B) where: (1) X is a v-element set of points; (2) B is a family of subsets of X called blocks; (3) K is a set of possible called block sizes; (4) If B ∈ B, then |B| ∈ K. (5) Every t-element subset of X is in exactly λ blocks. In Example 1.1 a 2-(7, 3, 1) design is exhibited. • If (1) t, v∈ / K, and X (2) k 6⊆ B, for any k ∈ K, then the tBD is said to be non-trivial ( or proper ). • If K = {k}, then the tBD is a t-design with parameters t-(v, k, λ). • A t-(v, K, 1) is called a Steiner system and the parameters are written as S(t, K, v). • An S(2, 3, v) is called a Steiner triple system and the notation ST S(v) is often used. The reverend T. Kirkman (4) has shown that a ST S(v) exists if and only if v ≡ 1, 3 (mod 6). • An S(3, 4, v) is called a Steiner quadruple system and the notation SQS(v) is often used. It was shown by Hanani (3) that a SQS(v) exists if and only if v ≡ 2, 4 (mod 6). 2.2. Orbits. If G is a subset of Sym(X), then G acts on the subsets of X in a natural way. If g ∈ Sym(X) and S ⊂ X, then g(S) = {g(x) : x ∈ S}. The orbit of S under the action of G is G(S) = {g(S) : g ∈ G}, and the stabilizer of S under the action of G is GS = {g ∈ G : g(S) = S}. The orbit counting lemma from group theory shows us that |G(S)| = |G|/|GS |. and the Cauchy-Frobenius-Burnside lemma tells us that the number of orbits of telement subsets under the action of G on X is 1 X Nt = χt (g), G g∈G where χt (g) = {T ∈ X : |T | = t, and g(T ) = T }. 5 2.3. Orbit incidence matrix. Let G ≤ Sym(X) be a possible automorphism group. Suppose t < k and let ∆ = G(T ) be the orbit of t-element subset T ⊆ X. Let Γ be an orbit of k-element subsets. B1 T B3 ∆= g Claim 1.2. 0 B2 0 g 0 |{B ∈ Γ : B ⊇ T }| = |{B ∈ Γ : B ⊇ T }| B10 T0 for all T ∈ ∆. B20 Orbit of t-element subsets. B30 This claim is easily verified. For if T 0 ∈ ∆, there is a g ∈ G such that T 0 = g(T ) and hence whenever T ⊂ B, then T 0 = g(T ) ⊂ Figure 1 g(B). This is illustrated in Figure 1. Claim 1.2 allows us to define the orbit incidence matrix. Given group G ≤ Sym(X) and 0 ≤ t ≤ k ≤ v = |X| let Orbit of k-element subsets. R = ∆1 , ∆2 , . . . , ∆Nt be the orbits of t-element subsets; C = Γ1 , Γ2 , . . . , ΓNk be the orbits of k-element subsets. The (t, k)-orbit incidence matrix is the matrix Atk : R × C → Z given by Atk [∆, Γ] = |{K ∈ Γ : K ⊇ T }|, where T ∈ ∆ is a fixed representative. An example is provided in Example 1.3. Example 1.3. I, (0, 3, 2)(1, 4, 6), (0, 2, 3)(1, 6, 4), Let G = h(0, 3, 2)(1, 4, 6), (0, 2, 3)(1, 6, 4)i = . (1, 4)(2, 3), (0, 2)(4, 6), (0, 3)(1, 6) The A2,3 matrix is: 046 016 134 014 136 346 246 146 126 124 036 045 026 356 123 125 013 034 126 015 056 035 024 236 145 345 135 235 234 012 023 456 256 245 025 {46, 16, 14} 36, 34, 26 12, 04, 01 {24, 13, 06} 1 1 2 0 0 0 1 0 0 0 0 1 1 1 1 0 0 1 0 0 0 0 2 2 0 0 0 0 1 0 {23, 03, 02} 0 0 0 2 1 1 0 0 0 1 {56, 45, 15} 0 0 0 0 0 0 2 2 1 0 {35, 25, 05} 0 0 0 0 0 0 0 2 1 2 =Γ 6 In 1976 Earl Kramer and Dale Mesner (7) made the following observation: A t-(v, k, λ) design exists with G ≤ Sym(X) as an automorphism group if and only if there is a (0,1)-solution U to the matrix equation Atk U = λJ, where: J = [1, 1, 1, . . . , 1]T . The orbits Γ, where U [Γ] = 1 form the design. In Example 1.3 U = [0, 1, 0, 0, 0, 1, 0, 0, 1, 0] is a solution to A23 U = J. The orbits of 3-element subsets corresponding the 1s in U from the design. In this case 056 014 135 346 ∪ 023 ∪ 245 126 forms a 2-(7, 3, 1) design. The very design displayed in Example 1.1. In order to cover t-wise balanced designs we extended our notion of orbit incidence by defining At∗ as At∗ = [At,t+1 , At,t+2 , At,t+3 , . . . At,v ] With this notation we see that: A t-(v, K, λ) design exists with G ≤ Sym(X) as an automorphism group if and only if there is a (0,1)-solution U to the matrix equation At∗ U = λJ, where: J = [1, 1, 1, . . . , 1]T . This observation on the orbit incidence matrices lead to following method for constructing t-designs. Step 1: Choose parameters t, k, v, and λ; Step 2: Find a candidate for an automorphism group G; Step 3: Generate the incidence matrix Atk ; Step 4: Solve the system of equations Atk U = λJ for one, some or all (0,1)vectors U ; Step 5: Check for any special properties you may require of the found solutions; Step 6: Apply any known recursive methods to the solutions found to construct more designs. Although each one of these steps poses interesting an challenging problems almost every known t-design with t ∈ {4, 5} was found this way. And every known t-design with t ≥ 6 was found this way. 7 3. Available ingredients The available combinatorial and algebraic structures have directed the possible research on t-wise balanced designs. Here is a rough classification. t = 2, 3: Latin squares, transversal designs, orthogonal arrays of strength 2 and 3, rich source of 2- and 3- homogeneous groups, recursive constructions, geometry, coding theory.. t = 4, 5: a few 4 and 5 homogeneous groups, union of orbits under other groups, coding theory. t ≥ 6: union of group orbits, Example 1.4. An example of using Latin Squares is the construction due to Bose of a STS(v) (X, B) when v ≡ 3 (mod 6). On Z2n+1 = {0, 1, 2, . . . , 2n}, define L by L[x, y] = (n + 1)(x + y) (mod 2n). Then L is an idempotent commutative Latin square. We take as the point set X = Z2n+1 × {0, 1, 2} and we take as the block set B the triples of the following two types. Type 1: {(x, 0), (x, 1), (x, 2)} for each x ∈ Z2n+1 Type 2: {(x, 0), (y, 0), (L[x, y], 1)}, {(x, 1), (y, 1), (L[x, y], 2)} and {(x, 2), (y, 2), (L[x, y], 0)} for each x, y ∈ Q, x 6= y. The Type 2 triples are well defined, because L[x, y] = L[y, x] and L[x, x] = x. There are 2n + 1 (3 + 6n)(2 + 6n) v(v − 1) 1 v 2n + 1 + 3 = = = 2 6 6 3 2 triples. Exactly the number of triples required to cover each pair only once, i.e. the right number for an STS(v). Thus it suffices to show that every pair is in one of the chosen triples. Consider any pair of different points (p, i), (q, j) ∈ X. • If p = q, then they are in the Type 1 triple {(p, 0), (p, 1), (p, 2)}. • If i = j, then the pair of points are in the Type 2 triple {(p, i), (q, i), (L[p, q], i + 1)}. • If p 6= q and i 6= j, then without loss j = i + 1 and (q, i), (p, j) are in the Type 2 triple {(p, i), (q, j), (r, k)}, where k = −i − j (mod 3) and r is defined by L[p, r] = q. Example 1.5. A subgroup G ≤ Sym(X) is said to be t-homogeneous on X if for any t-element subsets S and T there is a g ∈ G such that g(S) = T . Thus if G is t-homogeneous on X and K ⊆ X is and any k-element subset, t < k < v = |X|, then the orbit Γ = G(K) is a t-(v, k, λ) design, λ = |G|/(|Gk | kt ). On well known 3-homogenous family of groups are the Linear fractional groups. Let Fq be the finite field of order q and let X = Fq ∪ {∞} (the so-called projective line). A mapping f : X → X of the form ax + b x 7→ cx + d 1 ∞ where a, b, c, d ∈ Fq , ∞ = 0, 1 = ∞, 1 − ∞ = ∞, ∞ − 1 = ∞ and ∞ = 1 is called a ∞ linear fractional transformation. The determinant of f is det f = ad − bc The set of all linear fractional transformations whose determinant is a non-zero square is LF(2, q), the linear fractional group. When q ≡ 1 (mod 4) it is not difficult to prove that LF(2, q) is 3-homogeneous on X. Hence every orbit of k-element subsets k > t is 8 a 3-(q + 1, k, λ) design for some λ. Incidentally LF(2, q) is isomorphic to PSL(2, q) the projective special linear group. According to Wikipedia, the free encyclopedia: Daniel Gorenstein announced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of the classification of quasithin groups. The completed proof of the classification was announced by Aschbacher (2004) after Aschbacher and Smith published a 1221 page proof for the missing quasithin case. In particular all t-homogenous groups were known by the early 1980s they are the 2-homogenous groups: SAF(q) = {x 7→ α2 x + β : α, β ∈ Fq } 2-transitive groups: AF(q) = {x 7→ αx + β : α, β ∈ Fq } 3-homogenous groups: LF(2, q) ∼ = PSL(2, q) acting on the projective line, q 6≡ 1 (mod 4). 3-transitive groups: GL(2, q) acting on the projective line. 4-homogenous groups: PΓL(2, 16) and PΓL(2, 31) both acting on the projective line 4-transitive groups: The Matheiu groups: M11 and M23 acting on 11 and 23 points respectively 5-transitive groups: The Matheiu groups: M12 and M24 acting on 12 and 24 points respectively In particular there are no t-homogenous groups when t > 5, thus if we are to construct a 6-wise balanced design, then necessarily it will be constructed by taking a union of group orbits. This group may of course be possibly be trivial. Because of the lack of t-homogeneous groups with t > 5 a group theorist who shall remain nameless foolishly claimed that “there will not be any 6-designs.” However the situation is more complicated and prudently Peter Cameron & Jack van Lint (1) wrote in there book Graphs, Codes and Designs, LMS Lecture Note Series 43, 1980 on page 1. The existence of non-trivial t–designs with t > 5 is the most important unsolved problem in the area. Then • Leavit and Magliveras (8) in 1982 construct a 6-(33, 8, 36) design using a union of orbits from PΓL(2, 32). • Kramer, Leavit and Magliveras (5) in 1984 construct a 6-(20, 9, 112) Kramer, Leavit, Magliveras, 1984 using a union of orbits from PSL(2, 19). and • Kreher and Radziszowski (6) in 1986 construct a 6-(14, 7, 4) design using a cyclic group of order 13. (This is the smallest possible 6-design that can exist.) • Teirlinck (9) in 1987 constructed a t-(v, t + 1, (t + 1)!2t+1 ) for all t and v ≡ t (mod t + 1)!2t+1 ) and v ≥ t + 1. Thus showing that t-designs exist for t. This still did not satisfy the nameless group theorist who revised their remark to be “there will not be any interesting Steiner 6-designs.” and Cameron & van Lint (2) prudently revised their book Graphs, Codes and Designs now LMS Student Texts 22, 1991 and write on page 2 9 The existence of Steiner systems with large t is possibly the most important problem in design theory. To date no Steiner 6-wise balanced design is known to exist. Research problem 1. Construct a t-(v, K, 1) design with t ≥ 6. In the next chapter we will investigate a class of designs that I at least find interesting. This investigate will illustrate will rely on the taking a union of group orbits. Research problem 2. Find or classify all t-wise balanced designs. (Well, at least the interesting ones.) 4. References (1) P. Cameron and J. van Lint, Graphs, Codes and Designs, LMS Lecture Note Series 43, 1980. (2) P. Cameron and J. van Lint Graphs, Codes and Designs, LMS Student Texts 22, 1991. (3) H. Hanani, On quadruple systems, Canad. J. Math. 12 (1960), 145–157. (4) T. P. Kirkman, On a problem in combinatorics, Cambridge Dublin Math. J. 2 (1847), 191-204. (5) E.S. Kramer, D.W. Leavitt, and S.S. Magliveras, Construction Procedures dor t-designs, Ann. Discrete Math. 15 (1976), 263–296. (6) D.L. Kreher and S.P. Radziszowski, The existence of Simple 6-(14, 7, 4) designs, Journal Combinatorial Theory, series A,: 43 (1986), 237–243. (7) E.S. Kramer and D.M. Mesner,: t-Designs on Hypergraphs, Discrete Math. 15 (1976), 263–296. (8) S.S. Magliveras and D.W. Leavitt, Simple 6-(33, 8, 36) designs from PΓL2 (32), in Comput. Group Theory, Proceedings, London Math. Soc. Sympos. Comput. Group Theory, pp. 337–352, Academic Press, New York, 1984. (9) L. Teirlinck, Non-trivial t-designs without repeated blocks exists for all t, Discrete Math. 65 (1987) 301-311. 10 Part 2. Graphical Designs 5. Introduction A graphical design is a proper t-wise balanced designs (X, B) with parameters t , K, λ that has the symmetric group Sn as an automorphism group. Thus X will be the set of v = n2 labelled edges of the undirected complete graph Kn with vertex set {1, 2, . . . , n} and and blocks are subgraphs. Moreover if B ∈ B, then all subgraphs isomorphic to B are also in B. See Figures 2 and 3. n 2 6 K = 1 5 6 B = 2 4 5 2 3 4 X = 1 12, 13, 14, 15, 16, 23, 24, 25, 26, 34, 35, 36, 45, 46, 56 = Points are edges! 3 15, 16, 56, 24 Blocks are subgraphs! Figure 2 6. Orbits of Sn acting on E(Kn ). 1 1 2 The required condition: if B ∈ B, then all subgraphs isomorphic to B are also in B means that the set of blocks in a graphical design is a union of isomorphism classes, i.e. orbits under the action of Sn on E(Kn ). Such orbits are completely described by providing a picture of an unlabeled graph. This is illustrated in Figure 3. 3 6 2 5 3 4 1 6 2 5 3 4 3 2 5 3 3 2 5 3 4 2 5 3 5 1 6 2 5 3 4 , 6 4 6 5 4 Figure 3. The orbit It is easy to see by carefully examining Figure 3 that the h i and the , -entry of A2 ∗ is 2 . 5 1 6 1 6 h 6 4 4 1 3 5 1 6 4 2 2 4 1 2 1 6 i . -entry of A2 ∗ is 1 11 Now that we have pictures for orbits it is easy to construct the orbit incidence matrix. For example here is the A2∗ orbit incidence matrix for S4 acting on E(K4 ). 0 0 4 2 4 4 1 1 2 1 5 4 We see (for example) from this matrix that the subgroups isomorphic to together form a 2-(6, {3, 4}, 2) design. and 7. Steiner graphical designs. We will now focus on the construction of Steiner graphical tBDs. If we were to proceed systematically we would first find the orbits that contain each orbit of telement subsets at most once. Then among these orbits we would seek a collection that contains all the t-sets each exactly once. Example 2.1. The unique graphical Steiner 1-wise balanced designs. 1 So, 4 6 2 2 2 8 5 gives a 1-(6, 2, 1) design. In Example 2.2 we provide a few of the columns of the A2∗ orbit incidence matrix of S6 acting on E(K6 ). Example 2.2. The two graphical Steiner 2-wise balanced designs. So, design. 1 3 1 0 1 ··· 2 1 0 1 0 ··· gives a 2-(15, 3, 1) design, and gives a 2-(15, {3, 5}, 1) Example 2.3. The unique graphical Steiner 3-wise balanced design. 12 So, 1 0 0 0 1 0 0 0 1 0 1 0 gives a 3-(10, 4, 1) design. Example 2.4. The unique graphical Steiner 4-wise balanced design. So, 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 gives a 4-(15, {5, 7}, 1) design. 13 Theorem 2.5. (Chouinard,Kramer,Kreher (2) 1983.) The only graphical Steiner t-wise balanced designs are on this list. 1-(6, 2, 1) n=4 2-(15, 3, 1) n=6 2-(15, {3, 5}, 1) n = 6 3-(10, 4, 1) n=5 4-(15, {5, 7}, 1) n = 6 To prove Theorem 2.5 suppose (X, B) is a non-trivial graphical t-( then • • • • n 2 , K, 1) design, No block has size t. No block is X = E(Kn ), i.e., no block is complete. B does not contain all the k-subsets for any 0 < k < v. If g ∈ Sn and B ∈ B, then g(B) ∈ B. Let V (Kn ) = {1, 2, . . . , n}. The proof is given in a series of Lemmas. The first lemma although simple is key. Lemma 2.6. Let g ∈ Sym(V ) and B ∈ B. Then |B ∩ g(B)| ≥ t ⇒ B = g(B). Proof. If |B ∩g(B)| ≥ t, then there is a t-element subset T ⊆ E(Kn ) that is contained in the block B and also in the block g(B). But (X, B) is a Steiner tBD and so B = g(B). g B: t g(B) : We use Lemma 2.6 to prove the next few lemmas by taking an appropriate choice for a t-edge subgraph T . The general strategy is when given T to consider the unique block that contains it. Because (X, B) is non-trivial B 6= T . Therefore there is a edge e ∈ B \ T . Now if g ∈ Sym(V ) is such that |g(B) ∩ B| ≥ t, it follows from Lemma 2.6 that g(e) ∈ B. Lemma 2.7. n 6= t + 1 14 Proof. Suppose n = t+1 and take T = K1,n−1 . We now use Lemma 2.6 to force the unique block B containing T to be complete. This 1 2 will be a contradiction because (X, B) is nonn 3 trivial. Without loss of generality we may T = assume that vertex n has valency n − 1 in .. . T and because B 6= T , B contains an edge n−2 n−1 {i, j} where 1 ≤ i < j < n. The subgroup of Sym(V ) fixing n is 2-transitive on {1, 2, . . . , n − 1}. So for any edge {i0 , j 0 } where 1 ≤ i0 < j 0 < n There is a permutation g such that g fixes the edges of T and g({i, j}) = {i0 , j 0 }. Therefor by Lemma 2.6 {i0 , j 0 } ∈ g(B) = B. Therefore B is complete, which as mentioned earlier is a contradiction. Therefore n 6= t + 1. Lemma 2.8. n ≤ 2t + 2 Proof. Suppose n ≥ 2t+3 and take T to be a tmatching. Use Lemma 2.6 to force the unique block B containing T to be complete. To see t 1 2 3 this consider the adjacent figure an let e be an edge in the block B that is not in T . Up to ··· isomorphism there are 3 possibilities for e. e = {1, 2}: If B contains such an edge, t+1 t+2 t+3 2t then t > 1. Thus because the au- T = tomorphisms of T are 2-transitive on n 2t+1 2t+2 ··· the vertices of valency 1 it follows from Lemma 2.6 that B contains K2t on the vertices {1, 2, .., 2t}. Now the permu≥3 tation g = (1, 2t + 1) fixes the 2t−1 = 2 (t−1)(2t−1) ≥ t edges of the K2t−1 on {2, 3, . . . , 2t} and thus by Lemma 2.6 B contains the K2t+1 on the vertices {1, 2, .., 2t, 2t + 1}. Now considering the permutation (2, 2t + 2)(t + 2, 2t + 3) we force the edge {2t + 1, 2t + 2} into B. Finally permuting the isolated vertices of T forces B to be complete. A contradiction. e = {1, 2t + 1}: If b contains this edge, then using Lemma 2.6 and the permutations (1, t + 1) and (2t + 1, 2t + 2, . . . , n) We obtain a K1,n−2t+1 + {t + 1, 2t + 1} on {1, t + 1, 2t + 1, 2t + 2, . . . , n − 2t}. The image of B under any permutation on {t + 1, 2t + 1, 2t + 2, . . . , n} intersects B in at least t edges. Thus B contains the complete graph on {t + 1, 2t + 1, 2t + 2, . . . , n}. Now g = (2, t + 1) is such that g(B) ∩ B ≥ t, because there are at least 2 edges on B among the vertices 2t + 1, 2t + 2, . . . , n. Therefore B contains the edge {1, 2} and we reach a contradiction via the previous case. e = {2t + 1, 2t + 2}: In this case we first permute the isolated points of T and thereby show that B is complete on the vertices 2t + 1, 2t + 2, . . . , n. Now the permutation (2, t + 1) fixes t edges and So {1, 2} = g({1, t + 1}) ∈ B. We reach a contradiction via the first case. 15 Lemma 2.9. If t ≥ 3, then n ≤ t + 2. Proof. Suppose n ≥ t + 3. we again use the general strategy, but have left the details for the reader. ................................................................................... t = 3, n ≥ 6: Lemma 3 ⇒ n ∈ {6, 7, 8}. Take T to be a 3-matching with n − ··· T = 6 isolated points. Use Lemma 2.6 to n −6≥t−3 force the unique block B containing T . . . . . to . . .be . . .complete. ........................................................................ t = 4, n ≥ 7: Take T to be a triangle with a pendant edge and n − 4 iso··· T = lated points. Use Lemma 2.6 to force n −4≥3 the unique block B containing T to be . . . . . complete. .............................................................................. t ≥ 5, n ≥ t + 3: Take T to be the cy··· cle Ct with n−t isolated points. Use T = Lemma 2.6 to force the unique block n−t≥3 . . . . .B . . .containing . . . . . . . . . . .T. . to . . .be . . . complete. ........................................................ Lemma 2.10. The only graphical t-(v, K, 1) designs with t ≤ 4 are on list given below. 1-(6, 2, 1) n=4 2-(15, 3, 1) n=6 2-(15, {3, 5}, 1) n = 6 3-(10, 4, 1) n=5 4-(15, {5, 7}, 1) n = 6 Proof. Define Nt ⊆ N by n ∈ Nt if there exist a graphical t-BD on Kn . Lemmas 3,4,5 and 6 ⇒ N1 ⊆ {3, 4}, N2 ⊆ {4, 5, 6}, N3 ⊆ {5}, N4 ⊆ {6}. Checking for (0, 1)-solutions U to [At,t+1 , At,t+2 , . . . , At,n ]U = J, for n ∈ Nt Only the designs on the above list are found. 16 Lemma 2.11. If t ≥ 5, then n 6= t + 2. Proof. Suppose t ≥ 5, then Ct−2 is a cycle with t − 2 distinct edges. Attempts to cover ˙ 2 ∪K ˙ 2 , the (t−2)-cycle and two disjoint edges, with a block the t-edge subgraph Ct−2 ∪K B easily force B to be complete. This is of course the usual contradiction. Suppose that (X, B) is any t-(v, K, λ) design and consider a subset S ⊂ X, where s = |S| < t. Define (X 0 , B 0 ) by X0 = X \ S B 0 = {B \ S : S ⊆ B ∈ B} If T 0 ⊆ X 0 , has size |T 0 | = n − s, then T = T 0 ∪ S has size t. Therefore T is contained in λ blocks B1 , B2 , . . . , Bλ ∈ B and hence T 0 is contained in the the λ blocks B1 \ S, B2 , \S . . . , Bλ \ S ∈ B 0 . Thus (X 0 , B 0 ) is a (t − s)-(v − s, K0 , λ) design called the derived design with respect to S. The next lemma provides a situation when the derived with respect to a subgraph of a non-trivial graphical design is again a non-trivial graphical design. Lemma 2.12. If n ≤ t and (X, B) is a non-trivial graphical t-BD, then the derived design with respect to a K1,n−1 is a non-trivial graphical t − n + 1-BD. Proof. Let (X 0 , B 0 ) be the derived design with respect to a fixed (labeled) K1,n . For example set S = {{n + 1, x} : 1 ≤ x ≤ n} ∼ = K1,n−1 X 0 = X \ S = E(Kn−1 ) n+1 1 2 3 n B = {B \ S : S ⊆ B ∈ B}. X0 Then it is easy to see that (X 0 , B) is graphical. If (X 0 , B 0 ) is trivial, then k−n+1 ⊆ n−1 0 B for some block size k of (x, B). Let w = 2 and count the t − n + 1-subsets of X 0 contained in these k − n + 1-element subsets in two ways to obtain: w k−n+1 w w−t+n−1 = ⇒ = 1. k−n+1 t−n+1 t−n+1 k−t Thus k − t = 0 ⇒ k = t or k − t = w − t + n − 1 ⇒ k = n2 = |X|. Contrary to (X, B) non-trivial. 0 Lemma 2.13. If t ≥ 5, then graphical t-wise balanced designs do not exist. Proof. If t ≥ 5, then n ≤ t by Lemmas 2.7 and 2.9. So Lemma 2.12 applies. Thus if (X, B) is a graphical t-BD with t ≥ 5, by infinite descent we may assume the derived design with respect to a K1,n is on the list. Therefore we need only consider parameters (t, n) where: (t, n) ∈ {(5, 5), (8, 7), (8, 6), (10, 7)}. t = 5, n = 5: Take T to be C5 . Use Lemma 2.6 to force the unique block B containing T to be complete. t = 8, n = 7: Take T to be K4 with two pendant edges and an isolated point. Use Lemma 2.6 to force the unique block B containing T to contain S = K1,6 . Then B \ S contains a triangle with a pendant edge. But there is no such block among the two designs with t = 8 − 6 = 2. 17 t = 8, n = 6: Take T to be K4 and two pendant edges. Use Lemma 2.6 to force the unique block B containing T to contain S = K1,5 . Then B \ S contains a triangle with a pendant edge. But there is no such block among the two designs with t = 8 − 5 = 3. t = 10, n = 7: Take T = Deriving with respect to S = leaves The only block in the 4-BD that contains this is Therefore the block B containing T contains Use Lemma 2.6 to force B to be complete. This completes the proof of Theorem 2.5. 18 8. Bigraphical designs A t–wise balanced design (X, B) of type t–(m · n, K, λ) is bigraphical if X is the set of edges of the complete bipartite graph Km,n and whenever B is a block and α is an automorphism of Km,n (that fixes the independent sets), then α(B) is also a block. Theorem 2.14. (Hoffman-Kreher (4) 1994) The bigraphical t-designs of index 1 are on this list: D1 : 2≤m≤n 1−(mn, n, 1) K1,n D2 : 2≤m≤n 1−(mn, m, 1) Km,1 D3 : D4 : D5 : D6 : D7 : D8 : D9 : m=n=2 1−(4, 2, 1) m=n=3 2−(9, 3, 1) m = 2, n = 4 3−(8, 4, 1) m = 2, n = 4 3−(8, 4, 1) m=n=4 3−(16, 4, 1) m=n=4 3−(16, {4, 6}, 1) m=n=4 5−(16, {6, 8}, 1) 8.1. Remarks on the 5–(16, {6, 8}, 1) design. 11 11 11 11 11 10 10 10 10 10 01 01 01 01 01 00 00 00 00 00 11 10 01 00 11 10 01 00 11 10 01 00 11 11 11 11 11 11 11 11 10 10 10 10 10 10 10 10 01 01 01 01 01 01 01 01 00 00 00 00 00 00 00 00 If we arbitrarily label the the vertices in the two independent sets of K4,4 with the vectors (0, 0), (1, 0), (0, 1), (0, 0) and if and edge joins vectors (a, b) and (c, d) we form 19 the 4-dimension vector (a, b, c, d). In this way we think of the edges as vectors in Z42 . This labeling gives a surprising alternative realization of the 5-(16, {6, 8}, 1) design. • The 8-element blocks are the 3–dimensional affine subspaces. • A 6-element set {x~1 , . . . , x~6 } is a block ⇐⇒ x~1 + · · · + x~6 = ~0. This vector space construction is due to R.M. Wilson (see (5)) and our labeling shows that our bigraphical design is isomorphic to Wilson’s original construction. 9. Multi-graphical designs Knr = Kn, n, n, . . . , n | {z } r When n = 1 theses are the graphical designs. There are 4 of them, with t ≥ 2. (Chouinard, Kramer and Kreher (2) 1983) When r = 2 theses are bigraphical designs. There are 7 of them, with t ≥ 2. (Hoffman and Kreher (4) 1994) When n > 1 and r > 2 there are 2 more. (Olsen and Kreher (7) 1998) n = 2, r = 3 2−(12, {3, 4}, 1) n = 2, r = 4 2−(24, {3, 4}, 1) Research problem 3. Investigate other interesting families of group actions and the t-wise balanced that can be constructed from them. For example consider the action of Sym(X) on the 3-element subsets of X, see (3). Chouinard (1) has shown that for any pair (t, λ) with t > 1 or λ odd, there cannot exist a non-trivial graphical t- n2 , K, λ design with n2t + λ + 4. Thus, in particular, for each such pair (t, λ) there are only a finite number of non-trivial graphical t-(v, K, λ) designs. He further shows that if we further assume no repeated blocks, then for all cases with t > 1 or λ 6= 2, there do not exist non-trivial graphical t- n2 , K, λ designs with n ≥ 2t + λ + 4. This suggest the following problem. Research problem 4. Consider a parameterized family of graphs Xi , i = 1, 2, 3, . . ., Let Gi = Aut(Xi ) the automorphism of Xi acting on E(Xi ) the edges of Xi and let C be the set of all t-wise balanced designs (E(Xi ), B) that have Gi as an automorphism group, for some i. Find necessary and sufficient conditions on t and λ (or just λ) for when |C| is finite. 10. References on graphical designs (1) L. G. Chouinard II, Bounding graphical t-wise balanced designs. Discrete Math. 159 (1996), 261–263. (2) L.G. Chouinard, D.L. Kreher and E.S. Kramer, Graphical t-wise Balanced Designs, Discrete Mathematics 46 (1983) 227–240. 20 (3) D. de Caen and D.L. Kreher, The 3-hypergraphical Steiner quadruple systems of order twenty, in Graphs, Matrices and Designs Ed. Rolf Rees, Lecture Notes in Pure and Applied Mathematics 139 (1992) 85–92. (4) D.G. Hoffman and D.L. Kreher, The Bigraphical t-Wise Balanced Designs of Index One, The Journal of Combinatorial Designs 2 (1994) 41–48. (5) E.S. Kramer, Some results on t-wise balanced designs, Ars Combin. 15 (1983), 179–192. (6) L.M. Weiss and D.L. Kreher, The Bigraphical t-Wise Balanced Designs of Index Two, The Journal of Combinatorial Designs 3 (1995) 233–255. (7) C.L. Olsen and D.L. Kreher, Steiner graphical t-wise balanced designs of type nr , Statistical Planning and Inference 86 (2000) 535–566. (8) Y.M. Chee, D.L. Kreher, Graphical Designs The CRC handbook of combinatorial designs C.J. Colbourn and J.H. Dinitz (Editors) CRC Press, Boca Raton, 2007. 21 Part 3. Incidence Algebra 11. Matrices and Relations In this chapter we will investigate the Algebra of orbit incidence matrices. We will think of a matrix M with rows labeled by a set R, columns labeled by a set C and entries in a set E as function M :R×C →E that is an E valued vector with coordinates R × C. We will call these R by C matrices and if E is a field then the set of all R by C is a vector space of dimension |R| · |C| over E. The entry of M with coordinates (r, c) ∈ R × C is denoted by M [r, c]. If M is a R by C matrix M and N is a C by S matrix then M N is the R by S matrix defined by X M N [r, s] = M [r, c]N [c, s] c∈C This is of course usual matrix multiplication. One interesting application of matrix multiplication is in counting incidence in the composition of relations. Consider the relation of adjacency for a graph (V, E). The adjacency matrix of a graph (V, E) is the V by V matrix 1 if x is adjacent to y; A[x, y] = 0 if not. A walk of length ` from vertex x to vertex y is a sequence of vertices x0 , x1 , x2 , . . . , x` where x = x0 , y = x` and xi−1 is adjacent to xi for i = 1, 2, . . . `. In elementary graph theory we prove by induction that the number of x to y walks of length ` is A` [x, y]. In general a relation from the set S to the set T is a subset of S ×T and we write x ρ y when (x, y) ∈ ρ. Given such a relation we define the S by T matrix Mρ : S × T → Q by 1 if x ρ y; Mρ [x, y] = 0 if not. For example if ρ is adjacency for the graph (V, E). Then S = T = V and Mρ = A the adjacency matrix. • If ρ ⊆ S × T and σ ⊆ T × U , then Mρ Mσ [x, y] = |{t ∈ T : x ρ t σ y}|. • If ρi ⊆ Ti × Ti+1 , i = 1, 2, . . . , `, then Mρ1 Mρ2 · · ·Mρ` [x, y]=|{(t1 , . . ., t`−1 )∈T1 ×· · ·×T`−1 :x ρ1 t1 ρ2 · · · t`−1 ρ` y}|. 12. Group actions Here is a quick review of permutation groups notation. • The symmetric group on the set Ω is denoted by Sym(Ω). • If G is a group and θ : G → Sym(Ω) is a homomorphism we say that G acts on Ω and write G|Ω. If θ is a monomorphism we say that the acton is faithful. • Given a group action G|Ω, if g ∈ Sym(X), and ω ∈ Ω, then ω g is the image of ω under θ(g), where θ is the group action homomorphism. • If G ≤ Sym(X), then G acts on the subsets of X as follows: S g = {g(x) : x ∈ S}, where S ⊆ X and g ∈ G. • If G|Ω and ω ∈ Ω, then 22 ω G = {ω g : g ∈ G} is the orbit of ω under (the action of) G; Gω = {g ∈ G : ω g = ω} is the stabilizer of ω in (the action of) G; Orbit counting lemma: |ω G | |Gω | = |G|. set of (G|Ω)-orbits is Ω/G. 1 X – Cauchy-Frobenius-Burnside Lemma: |Ω/G| = χ(g), |G| g∈G – – – • The where χ(g) = |{ω ∈ Ω : ω g = ω}|. • If G ≤ Sym(X), – G| Xt , the t-element subsets of X, t ≤ |X|. – G|P(X), the power set of X. 13. Algebra of G-invariant matrices. We will denote by MatQ (Ω) the vector space of Ω by Ω matrices with entries in the field Q the rationals If G acts on Ω, then G also acts on MatQ (Ω). If M ∈ MatQ (Ω) and g ∈ G, then the image of M under the action of g is M g [σ, τ ] = M [σ g , τ g ]. In particular if G ≤ Sym(X), then G acts on MatQ (P(X)). It is this action we are interested in. Here M g [S, T ] = M [S g , T g ], where S g = {g(x) : x ∈ S} and T g = {g(x) : x ∈ T }. The matrices AlgQ (G|X) = {M ∈ MatQ (P(X)) : M g = M }, form a matrix algebra we call the algebra of G-invariant matrices. To see that it is a sub-matrix algebra of MatQ (P(X)) it suffices to show that it is closed under product and linear combination. So suppose M, N ∈ AlgQ (G|X). Then for all g ∈ G, S, T ⊆ X, and α, β ∈ Q we have: (M N )g [S, T ] = M N [S g , T g ] X = M [S g , U ]N [U, T g ] U ⊆X X −1 −1 = M [S, U g ]N [U g , T ] U ⊆X X −1 −1 = M [S, U g ]N [U g , T ] = g −1 U X ⊆X M [S, U ]N [U, T ] = M N [S, T ] U ⊆X and (αM +βN )g [S, T ] = = = = = (αM +βN )[S g , T g ] (αM )[S g , T g ]+(βN )[S g , T g ] α(M [S g , T g ])+β(N [S g , T g ]) α(M g [S, T ])+β(N g [S, T ]) α(M [S, T ])+β(N [S, T ]) = (αM +βN )[S, T ] If M ∈ MatQ (P(X)/G), then Mtk is the projection of M onto the Xt /G × Xk /G coordinates. (This is just ordinary vector-space projection.) 23 g Any Xt /G by Xk /G matrix Mtk can be embedded as a matrix M tk in MatQ (P(X)) by padding with zeros. X /G k g M tk = X t /G 0 0 0 0 Mtk 0 0 0 0 P(X)/G P(X)/G Because f e ^ (M tk Nk` )t` = (M )tk (N )k` and f)tk + β(N e )tk ^ (αMtk + βNtk )tk = α(M g we simply write Mtk instead of M tk . There will be no confusion. 14. Special matrices There are three special matrices in MatQ (P(X)/G) that we will be interested in. These are defined by: A[∆, Γ] = |{K ∈ Γ : K ⊃ T0 }|, where T0 is a fixed representative of orbit ∆. Notice that Atk is our old friend the orbit incidence matrix. B[∆, Γ] = |{T ∈ ∆ : T ⊂ K0 }|, where K0 is a fixed representative of orbit Γ. |∆| if ∆ = Γ, D[∆, Γ] = . 0 if not To emphasize the dependency of A, B and D on the group action G|X we may write A(G|X), B(G|X) and D(G|X). We denote the trivial group {I} consisting of only the identity by I. We use W = A(I|X) = B(I|X), to denote this special matrix in honor of Rick Wilson (3) who studied it in 1982. Using the the notation of Section 11 we see that W = Mρ , where ρ is the relation on P(X) defined by subset inclusion, i.e. S ρ T if and only S ⊆ T . Thus 1 if T ⊆ K W [T, K] = . 0 if not Lemma 3.1. (Kreher (1) 1986) The Wilson matrix W ∈ AlgQ (G|X), for every subgroup G ≤ Sym(X). Proof. If T ⊆ K ⊆ X, then T g ⊂ K g , for all g ∈ G. Hence W g = W for all permutations g ∈ Sym(X) and therefore W ∈ AlgQ (G|X) for every subgroup G. Lemma 3.2. BD = DA. 24 Proof. Suppose G|X and let ∆, Γ ∈ P(X)/G. Then X (BD)[∆, Γ] = B[∆, Φ]D[Φ, Γ] = B[∆, Γ]|Γ| Φ∈P(X)/G = X |{T ∈ ∆ : T ⊆ K}| K∈Γ = XX W [T, K] = K∈Γ T ∈∆ = X XX W [T, K] T ∈∆ K∈Γ |{K ∈ Γ : K ⊇ T }| T ∈∆ X = |∆|A[∆, Γ] = D[∆, Φ]A[Φ, Γ] = DA[∆, Γ]. Φ∈P(X)/G 15. Fusion. The Fusion matrix for the group action G|X is the P(X) by P(X)/G matrix defined by ( 1 √ if S ∈ ∆ |∆| . F [S, ∆] = 0 if not Lemma 3.3. (Kreher (1) 1986) F F T is in the center of AlgQ (G|X). Proof. Let M ∈ AlgQ (G|X) and let S, T ∈ P(X). Then X M [S, U ] |T G | U ∈P(X) ∆∈P(X)/G U ∈T G 1 X 1 X M [S, T g ] 1 X = G M [S, U ] = G = M [S, T g ] |T | |T | g∈G |GT | |G| g∈G U ∈T G X X 1 1 X M [S g , T ] 1 −1 = M [S g , T ] = M [S g , T ] = G |G| g∈G |G| g∈G |S | g∈G |GS | X M [U, T ] 1 X = G M [U, T ] = |S | |S G | U ∈S G XU ∈S G X = F [S, ∆]F [U, ∆]M [U, T ] = (F F T M )[S, T ]. (M F F T )[S, T ]= X X M [S, U ]F [U, ∆]F [T, ∆] = ∆∈P(X)/G U ∈P(X) Lemma 3.4. (Kreher (1) 1986) F T F = I in MatQ (P(X)/G). Proof. Let ∆, Γ ∈ P(X)/G. Then F T F [∆, Γ] = X F [U, ∆]F [U, Γ]. U ∈P(X) Hence F T F [∆, Γ] = 0 if ∆ 6= Γ. Otherwise X X X 1 1 F T F [∆, ∆] = (F [U, ∆])2 = ( p )2 = =1 |∆| |∆| U ∈∆ U ∈∆ U ∈P(X) 25 16. The fundamental epimorphism. Using the fusion matrix of G|X we define a homomorphism τ of matrix algebras from AlgQ (G|X) to MatQ (P(X)/G), by τ (M ) = D−1/2 F T M F D1/2 Lemma 3.5. (Kreher (1) 1986) τ is an homomorphism. Proof. For all x, y ∈ Q and M, N ∈ AlgQ (G|X). It is easy to see that xτ (M ) + yτ (N ) = τ (xM + yN ), and by the Fusion Lemmas τ (M N ) = D−1 F T M N F D1/2 = D−1 F T M N F F T F D1/2 = D−1 F T M F F T N F D1/2 = D−1 F T M F D1/2 D−1/2 F T N F D1/2 = τ (M )τ (N ) Therefore τ is a homomorphism of algebras. Lemma 3.6. (Kreher (1) 1986) τ is an epimorphism. Proof. Given M ∈ MatQ (P(X)/G) define N ∈ AlgQ (G|X) by N [S, T ] = M [S G , T G ]/|T G |, for all S, T ⊆ X. Then if ∆, Γ ∈ P(X)/G we have τ (N )[∆, Γ] = (D−1/2 F T N F D1/2 )[∆, Γ] 1/2 X X |Γ| F [U, ∆]N [U, T ]F [T, Γ] = |∆| U ∈P(X) T ∈P(X) 1/2 X X |Γ| 1 1 = N [U, T ] 2 2 |∆| |∆| |Γ| U ∈∆ T ∈Γ 1 XX = N [U, T ] |∆| U ∈∆ T ∈Γ 1 X X M [∆, Γ] = = M [∆, Γ] |∆| U ∈∆ T ∈Γ |Γ| To emphasize its importance we call τ the fundamental epimorphism. Property 3.7. We have: τ (W ) = A and thus τ (Wtk ) = Atk . 26 Proof. Let ∆, Γ ∈ P(X)/G. Then 1/2 |Γ| τ (W )[∆, Γ] = (F T W F )[∆, Γ] |∆| 1/2 X X |Γ| = F [T, ∆]W [T, K]F [K, Γ] |∆| T ∈P(X) K∈P(X) 1 XX W [T, K] = |∆| T ∈∆ K∈Γ 1 X 1 X = |{K ∈ Γ : K ⊃ T }| = A[∆, Γ] = A[∆, Γ] |∆| T ∈∆ |∆| T ∈∆ Thus τ (Wtk ) = Atk by Section 13. T Property 3.8. We have: τ (W T ) = B T and thus τ (WtkT ) = Btk . Proof. Let Γ, ∆ ∈ P(X)/G. Then 1/2 |∆| T (F T W T F )[Γ, ∆] τ (W )[Γ, ∆] = |Γ| 1/2 X X |∆| = F [T, Γ]W T [T, K]F [K, ∆] |Γ| T ∈P(X) K∈P(X) 1 XX T W [T, K] = |Γ| T ∈Γ K∈∆ 1 X 1 X T = |{K ∈ ∆ : K ⊃ T }| = B [Γ, ∆] = B T [Γ, ∆] |Γ| T ∈Γ |Γ| T ∈Γ T Thus τ (WtkT ) = Btk by Section 13. Theorem 3.9. (Kreher (1) 1986) The map τ is an AlgQ (G|X) →MatQ (P(X)/G) epimorphism that carries equations in the Wtk and WtkT matrices to equations in the T Atk and Btk matrices. 17. Some applications Let |X| = v and suppose 0 ≤ t ≤ k ≤ ` ≤ v. X Wtk Wk,` [T, L] = K ∈ : T ⊆ K ⊂ L k `−t if T ⊆ L k−t = 0 if not `−t = Wt,` [T, L] k−t Therefore `−t Wtk Wk,` = Wt,` . k−t 27 Consequently by Theorem 3.9 Atk Ak,` = `−t At,` . k−t Let (X, B) be a t-(v, k, λ) design and let Wt,B be the Xt × B sub-matrix of W consisting only of those columns labeled by the subsets that are blocks in B. Wilson (3) in 1982 gave an inclusion-exclusion proof establishing. min e,f X T We,B Wf,B = T bie+f −i Wi,e Wif i=0 v−i−j k−i v−t k−t when e + f ≤ t and bij = λ . Consequently by Theorem 3.9 min e,f T Ae,B Bf,B = X T bie+f −i Bi,e Aif i=0 v−i−j when e + f ≤ t and bij = λ k−i v−t k−t . At e = f = ` ≤ t/2 this equation becomes: T A`,B B`,B = ` X T bi2`−i Bi,` Ai,` i=0 T this is a sum of positive semi-definite matrices bi2`−i Bi,` Ai,` and hence hasrank greater X T than the rank of any one of them. In particular Rank(b`` B`` A`` ) is /G. ` Theorem 3.10. (Kreher (1) 1986) A t-(v, k, λ) design (X, B) with G ≤ Sym(X) as an automorphism group, with t ≤ 2` and v ≥ k + ` satisfies X |B/G| ≥ /G ` The number of orbits of blocks is at least the number of orbits of `-element subsets. If t = 2 and ` = 1 and G = {I} the trivial group. we see that this theorem implies the number of blocks is at least the number points. This is Fisher’s inequality. A t-(v, k, λ) design is said to be block transitive if it has an automorphism group that is transitive on its blocks. Theorem 8 yields. Theorem 3.11. Every block transitive t-(v, k, λ) design is `-homogeneous for ` = bt/2c. There are no 6-homogeneous groups so therefore no block transitive 12-designs. A transitive group action G|X in which Gx has equal size orbits on X \ {x} is said to be 3 -transitive. . 2 The Theorem shows that a block transitive 3-designs is transitive on `-subsets for ` ≤ 32 . Of course ` in the theorem is integer, but can we prove something better. 28 Research problem 5. Prove my 1984 conjecture. Every block transitive 3-design is 3 -transitive. 2 18. References (1) D.L. Kreher, An Incidence Algebra for Combinatorial Designs with Automorphisms, Journal of Combinatorial Theory Series A 42 (1986) 239-251. (2) D.L. Kreher, A Generalization of Connors Inequality, Journal of Combinatorial Theory Series A 50 (1989) 259-268. (3) R.M. Wilson, Incidence Matrices of t-designs, Linear Algebra Appl. 46 (1982), 73-82. 29 Part 4. The hole size bound 19. Review and introduction Recall a t-wise balanced design (tBD) of type t-(v, K, λ) is a pair (X, B), where (1) X is a v-element set of points , (2) B is a collection of subsets of X called blocks , (3) B ∈ B ⇒ |B| ∈ K, and (4) every t-element subset of X is in exactly λ blocks. A tBD is proper if t < k < v for each k ∈ K and that it is Steiner if λ = 1. Example 4.1. A proper 2BD of type 2-(9, {3, 4}, 4). X={a, b, c, d, 1, 2, 3, 4, 5} abcd,a12,a13,a14,a15,a23,a24,a25,a34,a35,a45, abcd,b12,b13,b14,b15,b23,b24,b25,b34,b35,b45, B= abcd,c12,c13,c14,c15,c23,c24,c25,c34,c35,c45, abcd,d12,d13,d14,d15,d23,d24,d25,d34,d35, d45 B={xij : x ∈ {a, b, c, d}, i, j ∈ {1, 2, . . . , 5}} ∪{abcd, abcd, abcd, abcd} A 2BD of type 2-(9, {3, 4}, 4) 1 a 2 b c 3 4 d 5 Repeat 4 times. In this chapter we ask how big can a block be in a tBD. First we review the history of this problem. (1983) E.S. Kramer (2) obtained the following results: (a) If B is a block in a Steiner tBD, then |B| ≤ (v + t − 3)/2 for t ≥ 2. (b) If B is a block in a Steiner tBD, then |B| ≤ (v − 1)/2 for t = 2, 4 while |B| ≤ v/2 for t = 3, 5. (c) Conjecture: If B is a block in a Steiner tBD, then |B| ≤ (v − 1)/2 for t even while |B| ≤ v/2 for t odd. (2000) M. Ira and E.S. Kramer (4) prove: 30 (d) If B is a block in a Steiner 6BD, then |B| ≤ v/2. (2001) D.L. Kreher and R.S. Rees settle Kramer’s conjecture: (e) If B is a block in a Steiner tBD, then |B| ≤ (v − 1)/2 for t even while |B| ≤ v/2 for t odd. 20. Incomplete t-wise balanced designs. An incomplete t-wise balanced design (ItBD) of type t-(v, h, K, λ) is a triple (X, H, B), where • • • • • X is a v-element set of points, H is an h-element set of points called the hole, B is a collection of subsets of X called blocks , B ∈ B ⇒ |B| ∈ K, and every t-element subset of X is either in the hole or in exactly λ blocks, but not both. Example 4.2. A proper I2BD of type 2-(9, 4, {3}, 4). X={a, b, c, d, 1, 2, 3, 4, 5} H = {a, b, c, d} a12,a13,a14,a15,a23,a24,a25,a34,a35,a45, b12,b13,b14,b15,b23,b24,b25,b34,b35,b45, B= c12,c13,c14,c15,c23,c24,c25,c34,c35,c45, d12,d13,d14,d15,d23,d24,d25,d34,d35, d45 ={xij : x ∈ {a, b, c, d}, i, j ∈ {1, 2, . . . , 5}} An I2BD of type 2-(9, {3}, 4) 1 a 2 b c 3 4 d 5 The hole. Comparing Examples 4.1 and 4.2 it is not difficult to see that a ItBD of type t(v, h, K, λ) is equivalent to a tBD of type t-(v, K∪{h}, λ) with a block of size h repeated λ times. (More than equivalent they are identical.) Thus in particular a Steiner tBD is a ItBD with λ = 1 in which any block of the tBD can be considered as the hole. We now prove our main theorem: 31 Theorem 4.3. (Kreher-Rees (5)) If t ≥ 2 and H is a hole in a ItBD with any λ, then |H| ≤ (v − 1)/2 for t even while |H| ≤ v/2 for t odd. and obtain as corollary Corollary 4.4. If B is a block in a Steiner tBD, then |B| ≤ (v − 1)/2 for t even while |B| ≤ v/2 for t odd. which of course settles Kramer’s conjecture. We will now show that we need only consider proper ItBDs of type t-(v, h, {t+1}, λ). Theorem 4.5. Suppose there exists a proper ItBD of type t-(v, h, K, λ) with 2 ≤ t ≤ h < v. Then there exists a proper ItBD of type t-(v, h, {t + 1}, λ0 ) where Y (k − t). λ0 = λ k∈K Proof. Let (X, H, B) be a proper tBD of type t-(v, h, K, λ) with 2 ≤ t ≤ h < v. Let K = {k1 , k2 , . . . , k` }. For each i = 1, 2, . . . , ` and each block B ∈ B of size ki , take ci = ` Y (kj − t) j=1,j6=i copies of B and then construct on each copy the t-(ki , t + 1, ki − t) design obtained by taking all of the (t + 1)-element subsets of B. The result is an ItBD design of type t(v, h, {t+1}, λ0 ). To see this let T be any t-element subset of the point set X. If T ⊆ H, then T was not contained in any block in the original design and consequently T is not contained in any block in the new design. Otherwise, suppose that T is contained in ri blocks of size ki in the original design, i = 1, 2, . . . , `. Then r1 + r2 + · · · + r` = λ. Hence in the new design, the number of blocks that T is contained in is r1 c1 (k1 − t) + r2 c2 (k2 − t) + · · · + r` c` (k` − t) ( ) ` ` X Y = ri (kj − t) (ki − t) i=1 j=1,j6=i ` Y = (r1 + r2 + · · · + r` ) (kj − t) j=1 =λ Y (k − t) k∈K Ki T 1 Figure 4. T is in ki −t (t+ 1)-subsets of Ki . as required. (See figure 4.) Therefore, it we need only prove Theorem 4.3 when K = {t + 1}. If (X, H, B) is an incomplete t-(v, h, K, λ) design and x ∈ H, then the derived design with respect to x is the incomplete (t − 1)-(v − 1, h − 1, K − 1, λ) design, (X \ {x}, H \ {x}, B 0 ) where, B 0 = {B \ {x} : x ∈ B ∈ B}. 32 Our next Lemma uses the derived design to simplify the problem. (2): Lemma 4.6. If Theorem 4.3 is true when t is even, then it is also true when t is odd. Proof. Suppose to the contrary that we have a proper ItBD of type t-(v, h, K, λ) with h > v/2, where t is odd, t ≥ 3. Then v ≤ 2h − 1. Taking the derived design through a point in the hole a proper I(t − 1)BD of type (t − 1)-(v − 1, h − 1, K − 1, λ) is obtained with v − 1 ≤ (2h − 1) − 1 = 2h − 2 = 2(h − 1). This contradicts the hypothesis, because t − 1 is even. Now we need only consider the case t even. To begin we first prove Theorem 4.3 for t = 2 and K = {3}. Lemma 4.7. If there is an incomplete 2-(v, h, {3}, λ) design with h < v, then v ≥ 2h + 1, with equality occurring if and only if every block intersects the hole in exactly one point. Proof. Let (X, H, B) be the indicated incomplete 2-(v, h, {3}, λ) design, and let x ∈ H. Then the derived design with λ times. respect to x yields a λ-regular multix graph on the vertex set X \ H; taking derived designs over all x ∈ H yields a h v−h λh-regular multi-graph on X \ H. Now let v ∈ X \ H. Then for any v 0 ∈ X \ H X\H with v 0 6= v, the number of times the pair The hole H {v, v 0 } occurs in our multi-graph cannot exceed λ, for otherwise the pair {v, v 0 } would have appeared in more than λ triples in (X, H, B). Hence, λh ≤ λ(v − h − 1) and so v ≥ 2h + 1 as desired. The equality situation follows easily. Now suppose t0 ≥ 4 is even and that we have established the non-existence of proper It0 BDs of types t0 -(2h − 1, h, {t0 + 1), λ) and t0 -(2h, h, {t0 + 1), λ), and that we have established Theorem 4.3 for t = t0 − 2. Then there cannot exist a proper It0 BD of type t0 -(v, h, {t0 + 1), λ) for any v ≤ 2h − 2, for otherwise, then deriving through two points in the hole would produce a proper ItBD of type t-(v − 2, h − 2, {t + 1), λ), where v − 2 ≤ 2h − 4 = 2(h − 2), contrary to our assumption. Hence Theorem 4.3 holds for t0 . By inductive reasoning, starting with Lemma 4.7, we can summarize the above discussion as follows: Lemma 4.8. If there do not exist proper ItBDs of type t-(2h − 1, h, {t + 1}, λ) or t-(2h, h, {t + 1}, λ) for any λ and any 2 ≤ t ≤ h, where t is even, then Theorem 4.3 holds ( for all 2 ≤ t ≤ h < v, K and λ). We now establish the non-existence of ItBDs of the type given in Lemma 4.8, and thereby establish Theorem 4.3. We begin by proving two combinatorial identities. We use the notation h (∗) gt (h) = h(h − 1)(h − 2) · · · (h − t + 1) = t! . t (Note g0 (h) = 1, the empty product.) Lemma 4.9. For every even integer t ≥ 2, and h ≥ t, t h−1 1X j (−1) gj−1 (h)gt−j+1 (h − 1) = . t−1 t! j=1 33 Proof. t 1X (−1)j gj−1 (h)gt−j+1 (h − 1) t! j=1 " # t X 1 −gt (h − 1) + g1 (h)gt−1 (h − 1) + = (−1)j gj−1 (h)gt−j+1 (h − 1) t! j=3 t h−1 h 1X (−1)j gj−1 (h)gt−j+1 (h − 1) = − + + t! j=3 t t Now if t = 2, then the sum term is vacuous; otherwise by observing that gj−1 (h)gt−j+1 (h − 1) = [h(h − 1) · · · (h − j + 2)][(h − 1)(h − 2) · · · (h − t + j − 1)] = [h(h − 1)(h − 2) · · · (h − t + j − 1)][(h − 1) · · · (h − j + 2)] = gt−j+2 (h)gj−2 (h − 1), we see that for each j = 3, . . . , 2t + 1, the j and t − j + 3 terms in the sum cancel. (They have opposite sign, because t is even). Thus in either case the sum term is equal to 0, and we get t h−1 h h−1 1X j (−1) gj−1 (h)gt−j+1 (h − 1) = − + = t! j=1 t t t−1 as desired. Lemma 4.10. For every even integer t ≥ 2, and h ≥ t t X tgt (h) . (−1)j−1 gj−1 (h)gt−j+1 (h) = − 2h − t j=1 Proof. We proceed by induction on t. When t = 2 we get 2 X (−1)j−1 gj−1 (h)g2−j+1 (h) = h(h − 1) − h2 = − j=1 as required. Now define ft (h) = t X j=1 (−1)j−1 gj−1 (h)gt−j+1 (h) 2h(h − 1) , 2h − 2 34 and let t ≥ 4. Then ft (h) = t X (−1)j−1 gj−1 (h)gt−j+1 (h) j=1 = t−1 X (−1)j gj (h)gt−j (h) j=0 t−2 X = gt (h) + (−1)j gj (h)gt−j (h) − gt−1 (h)h j=1 = gt (h) − gt−1 (h)h + h2 t−2 X (−1)j gj−1 (h − 1)gt−j−1 (h − 1) j=1 2 = gt (h) − gt−1 (h)h − h t−2 X (−1)j−1 gj−1 (h − 1)gt−j−1 (h − 1) j=1 2 (†) = gt (h) − gt−1 (h)h − h ft−2 (h − 1) (t − 2)gt−2 (h − 1) = gt (h) − gt−1 (h)h + h2 2(h − 1) − (t − 2) (t − 2)gt−1 (h) = gt−1 (h)(h − t + 1) − gt−1 (h)h + h 2h − t (−t + 1)(2h − t) + h(t − 2) = gt−1 (h) 2h − t −t = gt−1 (h)(h − t + 1) 2h − t −t = gt (h) , 2h − t as desired. (The induction assumption was used in the step labeled (†).) Now suppose that we have a proper ItBD (X, H, B) of type t-(2h − 1, h, {t + 1}, λ) with h ≥ t. For each i = 0, 1, · · · , t − 1, let Bi = {B ∈ B : |B ∩ H| = i} X \H H h i j h−1 and Ti = {T ⊆ X : |T | = tand |T ∩ H| = i}. . If we set xj = |Bj |, then counting pairs (T, B), such Let ti = |Ti |. Then ti = hi h−1 t−i that T ⊂ X, |T | = t, and T ⊂ B ∈ B in two ways we see that the following equations hold. (t + 1 − i)xi + (i + 1)xi+1 = λti for i = 0, 1, · · · , t − 2, and 2xt−1 = λtt−1 . 35 The coefficient matrix of this system of equations is t+1 1 t 2 t−1 3 Bt,t+1 = .. .. . . 3 t−1 2 . Remark 4.11. The matrix Bt,t+1 is the same Bt,t+1 matrix we saw in Chapter 3, Sections 14 and 13, with group G = Sym(H) × Sym(X \ H). If we set ~x = [x0 , x1 , . . . , xt−1 ] and ~t = [t0 , t1 , . . . , tt−1 ], then because each t-element subset of X not contained in H must occur in λ blocks, the matrix equation Bt,t+1~x = λ~t must hold. The matrix Bt,t+1 is upper triangular with non-zero main diagonal. Thus the matrix Bt,t+1 is invertible, whence −1 −1 ~ ~x = Bt,t+1 (λ~t) = λBt,t+1 t. −1 Indexing the rows and columns of Bt,t+1 by 1, 2, . . . , t it can be readily verified that −1 Bt,t+1 [1, j] = (−1)j−1 1 j t+1 j for j = 1, 2, . . . , t. Therefore x0 t X = λ (−1)j−1 h h−1 j−1 t−j+1 j t+1 j j=1 t 1 h h−1 λ X j (−1) t = − t + 1 j=1 j−1 t−j+1 j−1 ( ) t λ 1X = − (−1)j gj−1 (h)gt−j+1 (h − 1) . t + 1 t! j=1 1 Thus, applying Lemma 4.9 we see that −λ h − 1 x0 = < 0. t+1 t−1 This is a contradiction, because x0 counts the number of blocks disjoint from the hole H and consequently cannot be negative. Therefore, no proper ItBD of type t(2h − 1, h, {t + 1}, λ) with h ≥ t can exist. Now consider a proper ItBD of type t-(2h, h, {t + 1}, λ) with h ≥ t . Using the same analysis as before, we arrive at the matrix equation Bt,t+1~x = λ~t, 36 except this time for i = 0, 1, . . . , t − 1. So x0 = = = h h ti = i t−i in this case we get t X 1 h h j−1 λ (−1) t+1 j−1 t−j+1 j j j=1 t h h λ X j−1 1 (−1) t t + 1 j=1 j−1 t−j+1 j−1 ( t ) X λ (−1)j−1 gj−1 (h)gt−j+1 (h) (t + 1)! j=1 Thus, applying Lemma 4.10 we see that tgt (h) −λt h λ − = < 0, x0 = (t + 1)! 2h − t (t + 1)(2h − t) t again a contradiction. Thus, no proper ItBD of type t-(2h, h, {t + 1}, λ) with h ≥ t can exist. We have therefore established the following: Theorem 4.12. There do not exist proper ItBDs of types t-(2h − 1, h, {t + 1}, λ) or t-(2h, h, {t + 1}, λ) for any λ and any 2 ≤ t ≤ h, when t is even. Hence, by Lemma 4.8 we have established Theorem 4.3 and so the validity of Kramer’s conjecture (Corollary 4.4) follows. In particular, setting K = {k} in Corollary 4.4 we obtain the following: Theorem 4.13. (Kreher-Rees (5)) A Steiner system S(t, k, v) has k ≤ v/2 when t is odd and k ≤ (v − 1)/2 when t is even. Then in Section 21 we will construct some families of designs meeting the bounds of Theorem 4.3. In particular, for each odd t ≥ 3 and each h ≥ t + 1 we will construct a h−1 ItBD of type t-(2h, h, {t + 1}, (2h − t)t! t ); by deriving through a point in the hole in these designs we will obtain for each event ≥ 2 and each h ≥ t + 1 a ItBD of type h t-(2h + 1, h, {t + 1}, (2h − t + 1)(t + 1)! t+1 ). We will then show that there cannot exist any proper ItBD of type t-(v, h, K, λ) meeting the bounds of Theorem 4.3 when min{k : k ∈ K} ≥ t + 2. 21. Bounds are sharp In this section we will show that for every t and every h ≥ t + 1 there exists a ItBD that meets the bound given by Theorem 4.3. Hence, these bounds are sharp! We require the following results. Lemma 4.14. Let M be the t by t matrix h−t+i h−i M [i, j] = 0 whose [i, j]-entry is given by if j = i, if j = i + 1, if j 6= i or i + 1, for 0 ≤ i, j ≤ t − 1, and where h ≥ t + 1. Then the (unique) solution ~v to the matrix equation M~v = w, ~ 37 where w ~ = [0, 0, . . . , 0, h − t + 1]T has vt−i = (−1)i−1 gi (h − t + i) gi (h − 1) for i = 1, 2, . . . , t, and where gi are the functions defined by Equation (∗). Proof. We proceed by induction on i. For clarity, we first explicitly write out the matrix equation M~v = w: ~ h−t h 0 v0 v1 h−t+1 h−1 0 v2 h−t+2 h−2 0 . .. .. = .. .. . . . . 0 h − 2 h − t + 2 v t−2 h−1 vt−1 h−t+1 We have (h − 1)vt−1 = h − t + 1, and so h−t+1 . h−1 Consequently, the result is established for i = 1. Suppose 2 ≤ i ≤ t. Then from the matrix equation we see that vt−1 = (h − i)vt−i + (h − t + i)vt−(i−1) = 0, and so by induction we have vt−i −(h − t + i) −(h − t + i) = vt−(i−1) = h−i h−i i−2 gi−1 (h − t + i − 1) (−1) . gi−1 (h − 1) Hence, vt−i = (−1)i−1 gi (h − t + i) , gi (h − 1) as desired. Lemma 4.15. Let J = [1, 1, . . . , 1]T and let ~v be the vector defined in Lemma 4.14. If t is odd, then gt (h − 1)(J + ~v ) is a non-negative integer column vector ~u (where ut−i = gt (h − 1)(1 + vt−i ), i = 1, 2, . . . , t) in which ut−i = 0 if and only if i = t − 1. Proof. For each i = 1, 2, . . . t, ut−i = gt (h − 1)(1 + vt−i ) i−1 gi (h − t + i) = gt (h − 1) 1 + (−1) . gi (h − 1) Thus the vector ~u is integral, because gi (h − 1) divides gt (h − 1). Now we must show that gi (h − t + i) (‡) 1 + (−1)i−1 ≥ 0. gi (h − 1) It suffices to consider only even i , 2 ≤ i ≤ t − 1. When i is even inequality (‡) equivalent to gi (h − 1) ≥ gi (h − t + i), which follows easily because i ≤ t − 1. Note in particular that there is equality in inequality (‡) if and only if i = t − 1. The result follows. 38 We can now present our constructions. ˙ H, ) Theorem 4.16. For each odd t ≥ 3and each h ≥ t + 1 there exists a ItBD (H ∪Y, h−1 of type t-(2h, h, t + 1, (2h − t)t! t ), with Sym(H) × Sym(Y ) as an automorphism group. ˙ , where H and Y are disjoint sets of Proof. The point set of the design is X = H ∪Y cardinality h. The hole is H. There are precisely t orbits ∆0 , ∆1 , . . . , ∆t−1 of t-element subsets that need to be covered. The orbit ∆i is the set of t-element subsets that intersect the hole H in exactly i points, i = 0, 1, . . . , t − 1. Similarly, there are precisely t orbits Γ0 , Γ1 , . . . , Γt−1 of possible blocks ((t + 1)-element subsets) that are available. The orbit Γj is the set of all (t + 1)-element subsets that intersect the hole in exactly h j points. Thus |Γj | = hj t+1−j . Now consider the matrix At,t+1 whose [i, j]-entry is |{B ∈ Γj : T ⊆ B}|, ˙ H, ) of type where T is any fixed representative of ∆i . Then there is a ItBD (H ∪Y, h−1 t-(2h, h, t + 1, (2h − t)t! t ) with Sym(H) × Sym(Y ) as an automorphism group if and only if there is a non-negative integer vector ~u such that h−1 At,t+1~u = (2h − t)t! J. t Remark 4.17. The matrix At,t+1 is the same At,t+1 matrix we saw in Chapter 3, Sections 14 and 13, with group G = Sym(H) × Sym(X \ H). Furthermore, At,t+1 is the matrix M appearing in Lemma 4.14. Let ~u be the vector given in Lemma 4.15. Applying Lemmas 4.14 and 4.15 we see that At,t+1~u = At,t+1 [gt (h − 1)(J + ~v )] = gt (h − 1)At,t+1 (J + ~v ) = gt (h − 1)(At,t+1 J + At,t+1~v ) 0 2h − t 2h − t 0 . .. + .. = gt (h − 1) . 2h − t 0 h−1 h−t+1 = gt (h − 1)(2h − t)J h−1 = (2h − t)t! J. t The result follows. Remark 4.18. From Lemma 4.15, it follows that the vector ~u has u1 = 0. This means ˙ H, ) constructed in Theorem 4.16, there are no blocks in the that in the design (H ∪Y, design that intersect H in exactly one point! That is, the block orbit Γ1 is not used. h−1 Hence, the t-element subsets in ∆0 are covered λ = (2h − t)t! t times by the u0 copies of the orbit Γ0 . (This is a tBD (Y, 0 ) of type t-(h, {t + 1}, λ). Note that ’ is just the set of all t + 1-element subsets of Y each repeated u0 = λ/(h − t) times.) We can remove this sub-design and replace it with λ copies of the block Y to obtain a ItBD h−1 ? of type t-(2h, h, {t + 1, h }, (2h − t)t! t ). This gives the surprising result that the bound of Theorem 4.3 can be achieved without all blocks having size t + 1. 39 Theorem 4.19. For each even t ≥ 2 and each h ≥ t + 1 there exists a ItBD of type h t-(2h + 1, h, t + 1, (2h − t + 1)(t + 1)! t+1 ). ˙ H, ) of type (t + 1)-(2(h + Proof. From Theorem 4.16, there is a I(t + 1)BD (H ∪Y, (h+1)−1 1), h + 1, t + 2, (2(h + 1) − (t + 1))(t + 1)! t+1 ). Take the derived design through a point x ∈ H to get the desired ItBD. Remark 4.20. Recall that Sym(H) × Sym(Y ) was a group of automorphisms of the design constructed in Theorem 4.16. If x is any point in the design, then its stabilizer in Sym(H) × Sym(Y ) is an automorphism group of the derived design through x. Hence the designs constructed in Theorem 4.19 have Sym(H \ {x}) × Sym(Y ) as a group of automorphisms. In Remark 4.18 we noted that the bound of Theorem 4.3 can be achieved without all blocks having t + 1 points (at least when t is odd). We now show that whether t is even or odd, the bound cannot be achieved if the smallest block has t + 2 points (that is, there must be blocks of size t + 1 present in order to meet the bound). Note that it suffices to prove this theorem for even t, for if t is odd and there exists a proper ItBD of type t-(2h, h, K, λ) with min{k : k ∈ K} ≥ t + 2, then the derived design through a point in the hole would be a I(t − 1)BD of type t − 1-(2h − 1, h − 1, K − 1, λ) (meeting the bound of Theorem 4.3) with minimum block size at least t + 1 = (t − 1) + 2. We use the same strategy here as above. First of all, if there exists a proper ItBD of type t-(2h + 1, h, K, λ) with min{k : k ∈ K} ≥ t + 2, then by the obvious generalization of Theorem 4.5, there exists a proper ItBD of type t-(2h + 1, h, {t + 2}, λ0 ) for some λ0 Q (in fact λ0 = λ k∈K k−t ); thus, we can restrict ourselves to the case K = {t + 2}. 2 That is, it suffices to prove the non-existence of a ItBD of type t-(2h + 1, h, {t + 2}, λ) for each even t. An induction proof similar to Lemma 4.10 will establish the following combinatorial identity. Lemma 4.21. For every even integer t ≥ 2, and h ≥ t t X (−1)j−1 jgj−1 (h)gt−j+1 (h + 1) = − j=1 tgt+1 (h + 1) . 2h − t Proof. We proceed by induction on t. When t = 2, we get 2 X (−1)j−1 jgj−1 (h)g2−j+1 (h + 1) = (h + 1)h − 2h(h + 1) j=1 = −(h + 1)h = −2(h + 1)h(h − 1) 2h − 2 as required. Now define ft (h) = t X j=1 (−1)j−1 jgj−1 (h)gt−j+1 (h + 1) 40 and let t ≥ 4. Then ft (h) = t X (−1)j−1 jgj−1 (h)gt−j+1 (h + 1) j=1 = t−1 X (−1)j (j + 1)gj (h)gt−j (h + 1) j=0 = gt (h + 1) + t−2 X (−1)j (j + 1)gj (h)gt−j (h + 1) − tgt−1 (h)(h + 1) j=1 = gt (h + 1) +h(h + 1) t−2 X (−1)j (j + 1)gj−1 (h − 1)gt−j−1 (h) − tgt−1 (h)(h + 1) j=1 = gt (h + 1) + h(h + 1) t−2 X (−1)j gj−1 (h − 1)gt−j−1 (h) j=1 t−2 X −h(h + 1) (−1)j−1 jgj−1 (h − 1)gt−j−1 (h) − tgt−1 (h)(h + 1). j=1 Now applying Lemma 4.9 we see that t−2 X (−1) gj−1 (h − 1)gt−j−1 (h) = (t − 2)! j j=1 = = = = h−1 − gt−2 (h) + gt−2 (h − 1) (t − 2) − 1 (t − 2)gt−3 (h − 1) − gt−2 (h) + gt−2 (h − 1) (t − 2 − h)gt−3 (h − 1) + gt−2 (h − 1) −gt−2 (h − 1) + gt−2 (h − 1) 0. Hence, by induction Equation (§) becomes ft (h) = gt (h + 1) − h(h + 1)ft−2 (h − 1) − tgt−1 (h)(h + 1) −(t − 2)gt−1 (h) − tgt−1 (h)(h + 1) = gt (h + 1) − h(h + 1) 2(h − 1) − (t − 2) 2h − t + (t − 2)h − (2h − t)t = gt (h + 1) 2h − t −t(h − t + 1) = gt (h + 1) 2h − t tgt+1 (h + 1) = − , 2h − t as desired. Now suppose that we have a proper ItBD of type t-(2h + 1, h, {t + 2}, λ) for t even and h ≥ t. Using the the analysis of Theorem 4.16 we conclude that there must be a non-negative integer vector ~x = [x0 , x1 , . . . , xt−1 ] satisfying the matrix equation A~x = λ~t 41 where ~t = [t0 , t1 , . . . , tt−1 ], ti = hi h+1 , and t−i t+2 1 · (t + 1) 22 2 t+1 3 2 ·t 2 2 t 3 · (t − 1) 42 2 ... ... ... A= 5 t−1 (t − 2) · 4 2 2 4 (t − 1) 2 ·3 3 2 . Hence, ~x = λA−1~t. Indexing the rows and columns of A−1 by 1, 2, 3, . . . , t, the first row of A−1 can easily be shown to be A−1 [1, j] = (−1)j−1 j t t+2 2 j−1 for j = 1, 2, . . . , t, and so applying Lemma 4.21 (in the anti-penultimate step) h+1 h t λ X j−1 t−j+1 x0 = t+2 (−1)j−1 j t 2 = = λ t! j−1 j=1 t+2 2 λ t X (−1)j−1 jgj−1 (h)gt−j+1 (h + 1) j=1 −tgt+1 (h + 1) 2h − t t! t+2 2 −2λtgt+1 (h + 1) = (t + 2)!(2h − t) < 0, and we have a contradiction to the requirement x0 ≥ 0. Hence, no ItBD of the given type can exist. To summarize, we have shown the following: Theorem 4.22. There do not exist proper ItBDs of types t-(2h + 1, h, K, λ) (t even) or t-(2h, h, K, λ) (t odd) for any λ and any 2 ≤ t ≤ h, with min{k : k ∈ K} ≥ t + 2. Kreher and Rees (6) proved that if h is the size of a hole in an incomplete t-wise balanced design of order v and index λ having minimum block size k ≥ t + 1, then v + (k − t)(t − 2) − 1 . k−t+1 They showed that when t = 2 or 3, this bound is sharp infinitely often in that for each h ≥ t and each k ≥ t + 1, (t, h, k) 6= (3, 3, 4), there exists an ItBD meeting the bound. In (1) G, et al. we show that this bound is sharp infinitely often for every t. More precisely they showed for each t ≥ 4 that there exists a constant Ct > 0 such that whenever (h − t)(k − t − 1) ≥ Ct there exists an ItBD meeting the bound for some λ = λ(t, h, k). They then describe an algorithm by which it appears that one can obtain a reasonable upper bound on Ct for any given value of t. (§) h≤ 42 This result on the sharpness of Bound § is not very satisfactory so we offer the following problem. Research problem 6. Can necessary and sufficient conditions be found so that Bound § is sharp? Remark 4.23. We conclude by pointing out that if there exists a proper ItBD of type t-(v, h, K, λ) and a proper tBD type t-(h, K, λ), then we can construct the tBD on the points of the hole in the ItBD to obtain a proper tBD of type t-(v, K, λ) having a tBD of type t-(h, K, λ) as a (proper) sub-design. The reverse construction also holds: just remove the blocks (but not the points) of the sub-design to obtain an incomplete tBD. Thus Theorem 4.3 yields the following result concerning the maximal size of a sub-design in a tBD of type t-(v, K, λ): Corollary 4.24. Suppose that there is a proper tBD of type t-(v, K, λ) containing a tBD of type t-(w, K, λ) as a proper sub-design. Then v ≥ 2w when t is odd, while v ≥ 2w + 1 when t is even. 22. References (1) I. Adamczak, D.L. Kreher, A.C.H. Ling and R.S. Rees, Further results on the maximum size of a hole in an incomplete t-wise balanced design with specified minimum block size, J. Combin. Des. 10 (2002), no. 4, 256âĂŞ281. (2) E.S. Kramer, Some results on t-wise balanced designs, Ars Combin. 15 (1983), 179–192. (3) E.S. Kramer, “t-wise balanced designs,” The Handbook of Combinatorial Designs, C.J. Colbourn and J.H. Dinitz (Editors), CRC Press, 1996. (4) M. Ira and E.S. Kramer, A block-size bound for Steiner 6-wise balanced designs, J. Combin. Designs, 8 (2000), 141-145. (5) D.L. Kreher and R.S. Rees, A hole-size bound for incomplete t-wise balanced designs. J. Combin. Des. 9 (2001) 269–284. (6) D.L. Kreher, R.S. Rees, On the maximum size of a hole in an incomplete twise balanced design with specified minimum block size, to appear in the OSU Mathematical Research Institute Monograph Series. Index acts, 4 adjacency, 21 adjacency matrix, 21 algebra of G-invariant matrices, 22 AlgQ (G|X), 22 projection, 22 Mtk , 22 g M tk , 23 projective line, 7 projective special linear group PSL(2, q), 8 proper, 4, 29 bigraphical , 18 block sizes, 4 block transitive, 27 blocks, 4, 29, 30 rationals, 22 Q, 22 relation Mρ , 21 matrix, 21 relations, 21 Cauchy-Frobenius-Burnside, 4, 22 derived design, 16, 31 determinant, 7 det f , 7 special matrices, 23 A, 23 B, 23 D, 23 W, 23, 24 stabilizer, 4 Steiner, 29 Steiner quadruple system, 4 S(3, 4, v), 4 SQS(v), 4 Steiner system, 4 S(t, K, v), 4 t-(v, K, 1), 4 Steiner triple system, 4 S(2, 3, v), 4 ST S(v), 4 symmetric group, 3, 21 Sym(X), 3 Sym(Ω), 21 Fano plane, 3 finite field, 7 Fq , 7 fundamental epimorphism, 25 τ , 25 Fusion matrix, 24 general strategy, 13 graphical design, 10 group action, 21 G|Ω, 21 3 2 -transitive, 27 faithful, 21 group action homomorphism, 21 hole, 30 incomplete t-wise balanced design, 30 t-(v, h, K, λ), 30 ItBD, 30 t-homogeneous, 7 t-wise balanced design, 4, 29 t-(v, K, λ), 4, 29 tBD, 4, 29 Kramer-Mesner Observation, 6 Latin Squares, 7 linear fractional group, 7 LF(2, q), 7 linear fractional transformation, 7 vector space of Ω by Ω matrices MatQ (Ω), 22 walk, 21 Wikipedia, 8 Wilson matrix, 23 matrix, 21 non-trivial, 4 orbit, 4 orbit counting lemma, 4 orbit incidence matrix, 5 At∗ , 6 Atk , 5 points, 4, 29, 30 power set, 22 P(X), 22 43

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