CEJM 2 (2003) 198{207
Standard monomials for q-uniform families and a
conjecture of Babai and Frankl
G¶abor Heged}
us2 , Lajos R¶onyai12¤
1
2
Computer and Automation Institute
Hungarian Academy of Sciences
H-1111 Budapest, L¶agym¶anyosi u. 11, Hungary
Budapest University of Technology and Economics
H-1111 Budapest, M^
uegyetem rkp. 3{9, Hungary
Received 28 October 2002; revised 11 March 2003
Abstract: Let n; k; ¬
q-uniform family
be integers, n; ¬
> 0, p be a prime and q = p . Consider the complete
F (k; q) = fK ³ [n] : jKj ² k (mod q)g:
We study certain inclusion matrices attached to F (k; q) over the ­ eld Fp . We show that if
` µ q ¡ 1 and 2` µ n then
µ
¶
µ ¶
[n]
n
rank p I(F (k; q);
)µ
:
µ`
`
This extends a theorem of Frankl [7] obtained for the case ¬ = 1. In the proof we use arguments
involving Grobner bases, standard monomials and reduction. As an application, we solve a
problem of Babai and Frankl related to the size of some L-intersecting families modulo q.
c Central European Science Journals. All rights reserved.
®
Keywords: Grobner basis, inclusion matrix, set family
MSC (2000): 05D05, 13P10, 05B20
1
Introduction
¤
Throughout the paper n will be a positive integer and [n] stands for the set f1; 2; : : : ; ng.
The family of all subsets of [n] is denoted by 2[n] . For an integer 0 µ d µ n we denote by
¡[n]¢
¡ [n]¢ ¡[n]¢
¡ ¢
the family of all d element subsets of [n], and ·d
= 0 [ : : : [ [n]
the subsets of
d
d
size at most d.
E-mail: [email protected]
G. Heged}
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199
Babai and Frankl conjectured the following in [3], p. 115.
Theorem 1.1. Let k be an integer and q = p , ¬ ¶ 1, a prime power. Suppose that
2(q ¡ 1) µ n. Assume that F = fA1 ; : : : ; Am g is a family of subsets of [n] such that
(a) jAi j ² k (mod q ) for i = 1; : : : ; m
(b) jAi \ Aj j 6² k (mod q ) for 1 µ i; j µ m; i 6= j:
Then
mµ
µ
q¡
n
1
¶
:
In the proof we combine the linear algebra bound method with an argument involving
GrÄobner-standard monomials and the corresponding reduction.
For families F; G ³ 2[n] the inclusion matrix I(F ; G ) is a (0,1) matrix of size jF j £ jG j
whose rows and columns are indexed by the elements of F and G , respectively. The entry
at position (F; G) is 1 if G ³ F and 0 otherwise (F 2 F ; G 2 G ).
Let p be a prime and k an integer. Let q = p , ¬ ¶ 1. Let
F(k; q) = fK ³ [n] : jK j ² k (mod q)g:
In Theorem 1:1 of [7] Frankl proved the following Theorem.
Theorem 1.2. Let p be a prime and k an integer. If ` µ p ¡ 1 and 2` µ n, then
µ ¶
µ ¶
[n]
n
rank p I (F(k; p);
)µ
:
µ`
`
Our proof of Theorem 1.1 relies on the following generalization of Theorem 1.2.
Theorem 1.3. Let p be a prime and k an integer. Let q = p > 1. If ` µ q ¡
2` µ n, then
µ ¶
µ ¶
[n]
n
rank p I (F (k; q);
)µ
:
µ`
`
1 and
We give an equivalent form of Theorem 1.3 in Section 2 with the aid of standard
monomials. The bound in Theorem 1.3 is sharp. We have equality here if ` µ k µ n ¡ `
or if k + ` + q µ n. Indeed, by a theorem of Wilson [11] (see also Corollary 3.1 in [10])
¡ ¢ ¡[n]¢
¡n ¢
the sub-matrix I ( [n]
;
)
has
rank
, where m = k if ` µ k µ n ¡ ` and m = k + q
·`
m
`
if k + q + ` µ n.
2
Preliminaries
2.1 Polynomials, GrÄobner bases, standard monomials, reduction
Let F be a ¯eld. As usual, F[x1 ; : : : ; xn ] denotes the ring of polynomials in variables
x1 ; : : : ; xn over F. Let S = F[x1 ; : : : ; xn ]. In this paper F will be a ¯nite prime ¯eld Fp or
200
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the ¯eld of rational numbers Q.
We denote by F[x1 ; : : : ; xn ]·s the vector space of all polynomials over F with degree
at most s.
Q
For a subset F ³ [n] we write xF = j2F xj . In particular, x; = 1.
We recall now some basic facts concerning GrÄobner bases in polynomial rings. A total
order ¿ on the monomials (words) composed from variables x1 ; x2 ; : : : ; xn is a term order,
if 1 is the minimal element of ¿ , and uw ¿ vw holds for any monomials u; v; w with u ¿ v.
There are many interesting term orders. For the rest of the paper we assume that the
term order ¿ we work with is the deglex order. Let u = xi11 xi22 ¢ ¢ ¢ xinn and v = xj11 xj22 ¢ ¢ ¢ xjnn
be two monomials. Then u is smaller than v with respect to deglex (u ¿ v in notation)
i® either deg u < deg v, or deg u = deg v and ik < jk holds for the smallest index k such
that ik 6= jk . Note that we have xn ¿ xn¡1 ¿ : : : ¿ x1 .
The leading monomial lm(f ) of a nonzero polynomial f 2 S is the largest (with
respect to ¿ ) monomial which appears with nonzero coe±cient in f when written as a
linear combination of monomials.
Let I be an ideal of S. A ¯nite subset G ³ I is a GrÄobner basis of I if for every f 2 I
there exists a g 2 G such that lm(g) divides lm(f ). In other words, the leading monomials
of the polynomials from G generate the semi-group ideal of monomials flm(f ) : f 2 Ig.
Using the fact that ¿ is a well founded order, it follows that G is actually a basis of I ,
i.e. G generates I as an ideal of S. It is a fundamental fact (cf. [6, Chapter 1, Corollary
3.12] or [1, Corollary 1.6.5, Theorem 1.9.1]) that every nonzero ideal I of S has a GrÄobner
basis.
A monomial w 2 S is called a standard monomial for I if it is not a leading monomial
of any f 2 I. Let sm(¿ ; I ; F) stand for the set of all standard monomials of I with respect
to the term-order ¿ over F. It follows from the de¯nition and existence of GrÄobner bases
(see [6, Chapter 1, Section 4]) that for a nonzero ideal I the set sm(¿ ; I ; F) is a basis of
the F-vector-space S=I . More precisely every g 2 S can be written uniquely as g = h + f
where f 2 I and h is a unique F-linear combination of monomials from sm(¿ ; I ; F).
Let vF 2 f0; 1gn denote the characteristic vector of a set F ³ [n]. For a family of
subsets F ³ 2[n] , let V (F) = fvF : F 2 F g ³ f0; 1gn ³ Fn . To obtain information on
polynomial functions on V (F ), it is natural to consider the ideal I (V (F )):
I (V (F)) := ff 2 S : f (v) = 0 whenever v 2 V (F)g:
If F ³ 2[n], then x2i ¡ xi 2 I (V (F)), hence x2i is a leading monomial for I (V (F )). It
follows that the standard monomials for this ideal are all square-free, i.e. of form xG for
G ³ [n]. We put
Sm(¿ ; F; F) = fG ³ [n] : xG 2 sm(¿ ; I (V (F )); F)g ³ 2[n] :
It is immediate that Sm(¿ ; F ; F) is a downward closed set system. Also, the standard
monomials for I (V (F)) form a basis of the functions from V (F ) to F (see Section 4 in [2]),
hence
jSm(¿ ; F ; F)j = jF j:
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201
It is also easy to see that if H ³ G are arbitrary set systems on [n], then Sm(¿ ; H; F) ³
Sm(¿ ; G ; F).
Now we introduce the notion of reduction. Let G be a set of polynomials in F[x1 ; : : : ; xn ]
and let f 2 F[x1 ; : : : ; xn ] be a ¯xed polynomial. We can reduce f by the set G with respect
to ¿ . This gives a new polynomial h 2 F[x1 ; : : : ; xn ].
Here reduction means that we possibly repeatedly replace monomials in f by smaller
ones (with respect to ¿ ) as follows: if w is a monomial occurring in f and lm(g) divides
w for some g 2 G (i.e. w = lm(g)u for some monomial u), then we replace w in f with
u(lm(g) ¡ g). Clearly the monomials in u(lm(g) ¡ g) are ¿ -smaller than w.
It is a fundamental fact that if G is a GrÄobner basis of I, then with G we can reduce
every polynomial into a linear combination of standard monomials for I . In particular,
f 2 I i® f can be G -reduced to 0.
Let I be an ideal of S = F[x1 ; : : : ; xn ]. The Hilbert function of the algebra S=I is the
sequence hS=I (0); hS=I (1); : : :. Here hS=I (m) is the dimension over F of the factor-space
F[x1 ; : : : ; xn ]·m =(I \ F[x1 ; : : : ; xn ]·m ) (see [5, Section 9.3]).
In the case when I = I (V (F )) for some set system F ³ 2[n], the number hF (m) :=
hS=I (m) is the dimension of the space of functions from V (F ) to F which can be represented as polynomials of degree at most m.
In the combinatorial literature this important quantity is expressed in terms of inclusion matrices. It is straightforward to verify that
µ
¶
[n]
hF (m) = rank I (F ;
):
(1)
µm
On the other hand, hS=I (m) is the number of standard monomials of degree at most
m with respect to an arbitrary degree-compatible term order, for instance deglex.
2.2 The polynomials f H;d
We introduce a family of polynomials with integer coe±cients. They played an important
role in the description of the GrÄobner bases for the complete uniform families given
in [10]. Let t be a integer, 0 < t µ n=2. We de¯ne Ht as the set of those subsets
H = fs1 < s2 < ¢ ¢ ¢ < st g of [n] for which t is the smallest index j with sj < 2j. Thus,
the elements of Ht are t-subsets of [n]. We have H 2 Ht i® s1 ¶ 2; : : : ; st¡1 ¶ 2t ¡ 2 and
st < 2t. It follows that st = 2t ¡ 1, and if t > 1, then st¡1 = 2t ¡ 2.
For the ¯rst few values of t it is easy to give Ht explicitly: we have H1 = ff1gg and
H2 = ff2; 3gg, and H3 = ff2; 4; 5g; f3; 4; 5gg.
For a subset J ³ [n] and an integer 0 µ i µ jJj we denote by ¼ J;i the i-th elementary
symmetric polynomial of the variables xj , j 2 J:
X
¼ J;i :=
xT 2 Z[x1 ; : : : ; xn ]:
T µJ;jT j=i
In particular, ¼
J;0
= 1.
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Now let 0 < t µ n=2, 0 µ d µ n and H 2 Ht . Put H 0 = H [ f2t; 2t + 1; : : : ; ng ³ [n].
We write
µ
¶
t
X
i
t¡i d ¡
fH;d = fH;d (x1 ; : : : ; xn ) :=
(¡ 1)
¼ H 0;i :
t¡ i
i=0
Speci¯cally, we have ff1g;d = x1 + x2 + ¢ ¢ ¢ + xn ¡
ff2;3g;d = ¼
U;2
¡
d, and
µ ¶
d
1)¼ U;1 +
;
2
(d ¡
where U = f2; 3; : : : ; ng.
Assume that 0 µ d µ n=2. We write
Md := ffs1 < : : : < sj g » [n] : j µ d and si ¶ 2i for 1 µ i µ jg:
(2)
In particular ; 2 Md . The sets Md were studied in [2], [8] and [10] in connection with
order shattering and GrÄobner bases for uniform families. In Theorem 1:4 and Lemma 2:3
of [2] it was proved that
µ ¶
n
jM d j =
d
holds for 0 µ d µ n=2.
The following statement is from [10]. We include a proof for the reader’s convenience.
Characteristic vectors are interpreted now as elements of Zn .
Proposition 2.1. Assume that 0 < t µ n=2, H 2 Ht and 0 µ d µ n.
(a) The degree of fH;d is t, lm(fH;d ) = xH , and the leading coe±cient is 1.
(b) If D ³ [n], jDj = d, then fH;d (vD ) = 0.
Proof. (a) From the de¯nition of fH;d it is immediate that deg fH;d µ t. The coe±cient
¡ ¢
of xH in fH;d is (¡ 1)t¡t d¡t
= 1, giving also that deg fH;d = t. Also, H is the lexicot¡t
graphically largest among the subsets of H 0 with at most t elements. This implies that
xH is the leading monomial of fH;d .
(b) Write v = vD . Recall that H 0 = H [ f2t; : : : ; ng has n ¡ t + 1 elements. From jDj = d
this gives that
jD \ H 0 j 2 fd; d ¡
1; : : : ; d ¡
t + 1g:
(3)
We have
fH;d (v) =
t
X
(¡ 1)
k=0
t¡k
µ
¶
d¡ k
¼
t¡ k
H 0;k (v)
=
t
X
(¡ 1)
k=0
t¡k
µ
d¡ k
t¡ k
¶µ
¶
jD \ H 0 j
:
k
We use the following identity involving binomial coe±cients
µ
x¡
d+t¡
t
1
¶
=
t
X
k=0
(¡ 1)
t¡k
µ ¶µ
¶
x
d¡ k
;
k
t¡ k
(4)
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203
valid for every x 2 C, d 2 Z and t 2 Z+ . From (4) we infer that
µ
¶
jD \ H 0 j ¡ d + t ¡ 1
fH;d (v) =
;
t
which is indeed 0 because of (3).
It remains to prove (4). We consider ¯rst the Vandermonde identity ([9], pp. 169-170)
µ
¶ X
¶
t µ ¶µ
x+s
x
s
=
;
(5)
t
k
t¡ k
k=0
which holds for all x; s 2 C and t 2 Z+ . By negating the upper index s on the right-hand
side we obtain
µ
¶ X
µ
¶
t µ ¶
x+s
x
s¡ k¡ 1
t¡k t ¡
=
(¡ 1)
:
¡ k
t
k
t
k=0
Finally the substitution s = t ¡
d¡
1 gives (4).
h
The next statement is a slight generalization of an argument from [10].
Proposition 2.2. Assume that 0 µ ` µ n=2 and let
G
`
= fg H : H 2 Ht ; 0 < t µ `g [ fx21 ¡
x1 ; : : : ; x2n ¡
xn g » F[x1 ; : : : ; xn ]
be a collection of polynomials such that the degree of gH is t and lm(gH ) = xH . Let
f 2 F[x1 ; : : : ; xn ] be a polynomial, deg f µ `, which is irreducible with respect to G ` and
¿ . Then f is an F-linear combination of monomials from
N
`
:= fxG : G 2 M` g » F[x1 ; : : : ; xn ]:
Remark. The term irreducible means here that no reduction of f by any element of G
is possible.
`
Proof. Assume for contradiction that f contains a monomial w not in N ` . We have
deg w µ `. Also, w is square-free; otherwise we could reduce it further by some binomial
x2i ¡ xi . We have therefore w = xU for some U = fu1 < : : : < um g with m µ `. The
condition U 2
= M` means now that there is an index i µ m µ ` such that ui < 2i. Let t
be the smallest such index i. Then by the de¯nition of t we have ui ¶ 2i for 1 µ i < t,
hence fu1 < : : : < utg := H 2 Ht . Also, we see that H ³ U . But then the leading term
xH of g H divides w, hence f is reducible with respect to G ` , a contradiction. This proves
the statement.
3
The main results
3.1 A generalization of Frankl’s Theorem
In view of (1) and the subsequent remark, Theorem 1.2 follows from the next statement.
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Theorem 3.1. Let ¿
and 2` µ n, then
be the deglex order, p be a prime and k an integer. If ` µ p ¡
1
µ
¶
µ ¶
[n]
n
jSm(¿ ; F (k; p); Fp) \
jµ
:
µ`
`
We give a generalization to q-uniform families. It is slightly stronger than Theorem 1.3.
Theorem 3.2. Let ¿ be the deglex order, p be a prime and q = p > 1. Suppose that
k; ` 2 N for which 0 µ k; ` < q, and 2` µ n. Then
µ
¶
[n]
³ M` ;
Sm(¿ ; F (k; q); Fp ) \
µ`
hence
µ
¶
µ ¶
[n]
n
jSm(¿ ; F (k; q); Fp) \
jµ
:
µ`
`
Proof. We intend to use Proposition 2.2. We exhibit a set of polynomials
G ` » Fp [x1 ; : : : ; xn ] satisfying the conditions of the Proposition, with the additional property that all elements of G ` vanish on the characteristic vectors of the family F(k; q).
This su±ces, because the standard monomials for V (F(k; q)) must be irreducible with
respect to any set of polynomials which vanish on the set, in particular, with respect to
G ` . On the other hand, the G ` -irreducible monomials are in N ` by Proposition 2.2. These
allow us to conclude that the standard monomials of degree at most ` for the complete
q-uniform family are in N ` .
We thus turn to the construction of G ` . Obviously the binomials x2i ¡ xi 2 Fp[x1 ; : : : ; xn ]
vanish on V (F (k; q)). For 0 < t µ ` and H 2 Ht we de¯ne gH 2 Fp [x1 ; : : : ; xn ] as the
modulo p reduction of the polynomial (with integer coe±cients) fH;k . By Proposition 2.1
(a) the degree of gH is t and the leading term of gH is xH .
It su±ces to verify that gH (vD ) = 0 for the characteristic vectors vD of elements
D 2 F (k; q). We recall the following simple fact.
Lemma 3.3. Let q = p > 1 a prime power. Let x; j be integers, 0 µ j < q. Then
µ
¶ µ ¶
x+q
x
²
(mod p):
j
j
Proof. The congruence follows from the Vandermonde identity (5) with s = q and t = j,
¡¢
by noting that the binomial coe±cients qi vanish modulo p for 1 µ i < q.
Now let D 2 F(k; q), and write v = vD . Then jDj = k 0 for some k 0 such that
0 µ k 0 µ n and k ² k 0 (mod q). We observe that fH;k ² fH;k 0 (mod p), i.e., the coe±cients
of the two polynomials are the same modulo p. This holds because for 0 µ i µ t we have
µ
¶ µ 0
¶
k¡ i
k ¡ i
²
(mod p);
t¡ i
t¡ i
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a consequence of 0 µ t ¡
We conclude that
iµq¡
205
1 and Lemma 3.3.
gH (v) ² fH;k (v) ² fH;k 0(v) = 0
(mod p):
Here the last equality follows from Lemma 2.1 (b). The proof is complete.
3.2 The conjecture of Babai and Frankl
We prove Theorem 1.1 by a combination of the linear algebra argument presented in
Theorem 5.30 of [3] with Theorem 3.2. We make ¯rst some preparations. The following
fact was proved in Proposition 5:31 of [3].
Proposition 3.4. Let q = p , p a prime, and ¬ ¶ 1. For any integer r, the binomial
¡ ¢
coe±cient r¡1
is divisible by p i® r is not divisible by q. h
q¡1
Let f (x1 ; : : : ; xn ) 2 Q[x1 ; : : : ; xn ] be a polynomial. The square-free reduction f 0 of f is
obtained by reducing f with respect to the set of polynomials fx21 ¡ x1 ; : : : ; x2n ¡ xn g. In
other words, we replace x2i with xi as long as it is possible. Clearly f 0 is a Q-linear combination of monomials xU , U ³ [n]. It is immediate that deg f 0 µ deg f and f (v) = f 0 (v)
for every vector v 2 f0; 1gn.
Lemma 3.5. Let f 2 Q[x1 ; : : : ; xn ] be a polynomial such that f (v) 2 Z for every
v 2 f0; 1gn . Let f 0 be the square-free reduction of f . Then f 0 2 Z[x1 ; : : : ; xn ]:
Proof. We have
f 0 (x1 ; : : : ; xn ) =
X
¬
H
¢ xH ;
(6)
Hµ[n]
where ¬ H 2 Q. Suppose for contradiction that f 0 62 Z[x1 ; : : : ; xn ]. Then there exists
G ³ [n] such that ¬ G 2 Q n Z. Let K be minimal with respect to inclusion among
P
those subsets G. Obviously ¬ K 62 Z. Then f (vK ) = f 0 (vK ) =
Y µK ¬ Y xY (vK ) =
P
P
2 Z. Also Y ½K ¬ Y 2 Z, by the minimality of K . These imply that
Y ½K ¬ Y + ¬ KP
0
¬ K = f (vK ) ¡
Y ½K ¬ Y is also in Z, a contradiction. This proves the claim.
Proof (of Theorem 1.1). Let vi denote the characteristic vector of the set Ai . Let us
consider the polynomials
µ
¶
x ¢ vi ¡ k ¡ 1
fi (x1 ; : : : ; xn ) =
q¡ 1
in n rational variables x = (x1 ; : : : ; xn ) 2 Qn (i = 1; : : : ; m); where a ¢ b denotes the
scalar product in Qn . Denote by fi0 the square-free reduction of fi for i = 1; : : : ; m. Then
fi0 2 Z[x1 ; : : : ; xn ], because fi (v) 2 Z for each v 2 f0; 1gn , and hence Lemma 3.5 applies.
Let gi 2 Fp [x1 ; : : : ; xn ] denote the reduction of fi0 modulo p and hi 2 Fp[x1 ; : : : ; xn ] the
reduction of gi by a deglex GrÄobner basis for the ideal I := I(V (F (k; q))) of polynomials
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vanishing on V (F (k; q)) (actually reduction by G
fi (vj ) = fi0 (vj ) ² gi (vj ) ² hi (vj )
q¡1
su±ces here). Obviously we have
(mod p) for 1 µ j µ m:
(7)
Here the (¯rst) equality is valid for any 0,1-vector, while the second congruence holds
because Aj 2 F (k; q). Next note that by Proposition 3.4 the integer
µ
¶
jAi \ Aj j ¡ k ¡ 1
fi (vj ) =
q¡ 1
6 j. Then (7) implies that hi (vj ) will be 0 in Fp i® i 6= j. We
will be divisible by p i® i =
thus found that the m £ m matrix H = (hi (vj ))m
i;j=1 is a diagonal matrix over Fp with
no zeroes in the diagonal. From Proposition 2:7 of [3] (Determinant Criterion) it follows
that the polynomials h1 ; : : : ; hm are linearly independent over Fp.
Moreover, being reduced polynomials with respect to a GrÄobner basis, the hi are linear
combinations of standard monomials for I and deg hi µ q ¡ 1 because deg fi = q ¡ 1, and
the reductions (modulo p, and deglex) involved can not increase the degree. By Theorem
3.2 we infer that the linearly independent polynomials h1 ; : : : ; hm are in the Fp -space
spanned by N q¡1 , and hence
µ
¶
n
jF j = m µ jN q¡1 j =
;
q¡ 1
which was to be proved.
4
A concluding remark
¡[n]¢
Here we gave an upper bound on the rank of inclusion matrices I(F (k; q); ·`
) over Fp ,
where 0 µ ` µ q ¡ 1. In fact, for n su±ciently large, we determined the rank precisely.
It would be interesting to describe the set of standard monomials Sm(¿ ; F (k; q); Fp) in
a fashion similar to the uniform case given in [10].
¡ ¢
This could give bounds on the rank of inclusion matrices I (F(k; q); [n]
·` ) over Fp ,
where ` > q ¡ 1.
Acknowledgment
Research supported in part by OTKA and NWO-OTKA grants, and the EU-COE Grant
of MTA SZTAKI.
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