Exercise Sheet for ‘The existence of designs’
Peter Keevash
1. Let X = Z2t+1 × Z3 and T be the set of all triples in X of the form {(x, 0), (x, 1), (x, 2)} or
{(x, i), (y, i), ((x + y)/2, i + 1)} with x �= y. Show that T is a Steiner Triple System on X.
2. Let X = (Z2t × Z3 ) ∪ {∞}. Write xi = (x, i) for x ∈ Z2t , i ∈ Z3 . Let B be the set of all triples in X
of the form {00 , 01 , 02 }, {∞, 0j , tj+1 }, {0j , ij+1 , (−i)j+1 } for i = 1, . . . , t − 1, {tj , ij+1 , (1 − i)j+1 } for
i = 1, . . . , t. Let T = {a0 + B : a = 0, 1, . . . , t − 1}, where x + B = {x + b : b ∈ B}, ai + bj = (a + b)i+j
and x + ∞ = ∞. Show that T is a Steiner Triple System on X.
3. Show that for any finite set X there is a set T of triples in X such that every pair of X is in at
most one triple of T , and for every x ∈ X there are at most four choices of y ∈ X such that xy is not
in a triple of T .
4. Assuming the theorem that tridivisible typical graphs have triangle decompositions, show that
G(n, 1/2) whp has a set of edge-disjoint triangles covering all but n/4+o(n) edges. Can you generalise?
5. Assuming the nibble, apply a random greedy algorithm to show that for any q, r there is n0 such
that for n > n0 there is a q-graph G on n vertices such that every set of vertices of size r is contained
in one or two edges of G.
6. Suppose X is a set and f assigns weights in Q/Z to pairs from X such that f (ab)+f (bc)+f (ca) = 0
for any distinct a, b, c. Show that f (ab) + f (cd) = f (ac) + f (bd) for any distinct a, b, c, d, deduce that
there is some g : X → Q/Z such that f (ab) = g(a) + g(b) for all distinct a, b, and so 6g = 0. Show that
if n is 1 or 3 mod 6 then there is an assignment of integer weights to the triangles in Kn so that every
edge receives weight 1 (i.e. there is Φ ∈ ZK3 (Kn ) such that ∂2 Φ = Kn ); you may assume (or prove
using Gaussian Elimination / Hermite Normal Form) that this is equivalent to showing that there is
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no f : E(Kn ) → Q/Z such that f (ab) + f (bc) + f (ca) = 0 for any distinct a, b, c but ab f (ab) �= 0.