Open Access

A new Alon-Babai-Suzuki-type inequality on set systems

Journal of Inequalities and Applications20132013:171

https://doi.org/10.1186/1029-242X-2013-171

Received: 1 February 2013

Accepted: 2 April 2013

Published: 15 April 2013

Abstract

Let K = { k 1 , k 2 , , k r } and L = { l 1 , l 2 , , l s } be two sets of nonnegative integers. Let be a family of subsets of [ n ] such that | F | K for every F F and | E F | L for every pair of distinct subsets E , F F . We prove that

| F | ( n 1 s ) + ( n 1 s 1 ) + + ( n 1 s 2 r + 1 )

when we have the conditions that K L = and n s + max k i . This result gives an improvement for Alon, Babai, and Suzuki’s conjecture under the nonmodular. This paper gets an improvement of a theorem of Hwang et al. with K L = .

MSC:05D05.

Keywords

Alon-Babai-Suzuki inequalitiesFrankl-Ray-Chaudhuri-Wilson theoremsmultilinear polynomials

1 Introduction

In this paper, let X denote the set [ n ] = { 1 , 2 , , n } and L = { l 1 , l 2 , , l s } be a set of s nonnegative integers. A family of subsets of X is called -intersecting if | E F | L for every pair of distinct subsets E , F F . A family is k-uniform if it is a collection of k-subsets of X.

In 1975, Ray-Chaudhuri and Wilson proved the following fundamental result.

Theorem 1.1 (Ray-Chaudhuri and Wilson [1])

If is a k-uniform -intersecting family of subsets of X, then
| F | ( n s ) .

In terms of the parameters n and s, this inequality is best possible, as shown by the set of all s-subsets of X with L = { 0 , 1 , , s 1 } . A nonuniform version of Theorem 1.1 was proved by Frankl and Wilson in 1981.

Theorem 1.2 (Frankl and Wilson [2])

If is an -intersecting family of subsets of X, then
| F | ( n s ) + ( n s 1 ) + + ( n 0 ) .

This result is again best possible in terms of the parameters n and s, as shown by the family of all subsets of size at most s of X with L = { 0 , 1 , , s 1 } .

In 1991, Alon, Babai, and Suzuki proved the following inequality, which is a generalization of the well-known Frankl-Ray-Chaudhuri-Wilson theorems (Theorem 1.1 and Theorem 1.2).

Theorem 1.3 (Alon, Babai, and Suzuki [3])

Let K = { k 1 , k 2 , , k r } , L = { l 1 , l 2 , , l s } be two sets of nonnegative integers and let be an -intersecting family of subsets of X such that | F | K for every F F . If min k i > s r , then
| F | ( n s ) + ( n s 1 ) + + ( n s r + 1 ) .

Note that it is best possible in terms of the parameters n, r, and s, as shown by the set of all subsets of X with size at least s r + 1 and at most s, and L = { 0 , 1 , , s 1 } .

Since then, many Alon-Babai-Suzuki-type inequalities have been proved. Below is a list of related results in this field obtained by others.

Theorem 1.4 Let K = { k 1 , k 2 , , k r } , L = { l 1 , l 2 , , l s } be two sets of nonnegative integers and let be an -intersecting family of subsets of X such that | F | K for every F F . If K L = , then
| F | ( n 1 s ) + ( n 1 s 1 ) + + ( n 1 0 ) .

Theorem 1.5 (Snevily [4])

Let K = { k 1 , k 2 , , k r } , L = { l 1 , l 2 , , l s } be two sets of nonnegative integers and let be an -intersecting family of subsets of X such that | F | K for every F F . If min k i > max l j , then
| F | ( n 1 s ) + ( n 1 s 1 ) + + ( n 1 s 2 r + 1 ) .

Theorem 1.6 (Hwang and Sheikh [5])

Let K = { k , k + 1 , , k + r 1 } , L = { l 1 , l 2 , , l s } be two sets of nonnegative integers and let be an -intersecting family of subsets of X such that | F | K for every F F . If K L = and k > s r , then
| F | ( n 1 s ) + ( n 1 s 1 ) + + ( n 1 s r ) .

Theorem 1.7 (Hwang and Sheikh [5])

Let K = { k 1 , k 2 , , k r } , L = { l 1 , l 2 , , l s } be two sets of nonnegative integers and let be an -intersecting family of subsets of X such that | F | K for every F F . If F F F and min k i > s r , then
| F | ( n 1 s ) + ( n 1 s 1 ) + + ( n 1 s r ) .

Conjecture 1.8 (Alon, Babai, and Suzuki [3])

Let K and be subsets of { 0 , 1 , , p 1 } such that K L = , where p is a prime and F = { F 1 , F 2 , , F m } be a family of subsets of [ n ] such that | F i | ( mod p ) K for all F i F and | F i F j | ( mod p ) L for i j . If n s + max k i for every i, then | F | ( n s ) + ( n s 1 ) + + ( n s r + 1 ) .

In the following paper, they prove the above conjecture in the nonmodular.

Theorem 1.9 (Hwang et al. [6])

Let K = { k 1 , k 2 , , k r } , L = { l 1 , l 2 , , l s } be two sets of nonnegative integers and let be an -intersecting family of subsets of X such that | F | K for every F F . If n s + max k i , then
| F | ( n s ) + ( n s 1 ) + + ( n s r + 1 ) .

In this paper, we will prove the following Alon-Babai-Suzuki-type inequality which gives an improvement for Theorem 1.9 with K L = and for Alon, Babai and Suzuki’s conjecture in the nonmodular.

Theorem 1.10 Let K = { k 1 , k 2 , , k r } , L = { l 1 , l 2 , , l s } be two sets of nonnegative integers and let be an -intersecting family of subsets of X such that | F | K for every F F . If K L = and n s + max k i , then
| F | ( n 1 s ) + ( n 1 s 1 ) + + ( n 1 s 2 r + 1 ) .

We note that in some cases the conditions K L = and n s + max k i in Theorem 1.10 holds, but Snevily’s condition min k i > max l j in Theorem 1.5 does not. For instance, if n = 12 , K = { 3 , 6 } and L = { 0 , 1 , 2 , 4 , 5 } , then it is clear that K L = and s + max k i = 5 + 6 < 12 = n , but min k i = 3 < 5 = max l j .

If the condition K L = in Theorem 1.10 is replaced by F F F , we have the following result which gives a better bound than Theorem 1.10 and Theorem 1.7.

Theorem 1.11 Let K = { k 1 , k 2 , , k r } , L = { l 1 , l 2 , , l s } be two sets of nonnegative integers and let be an -intersecting family of subsets of X such that | F | K for every F F . If F F F and n s + max k i , then
| F | ( n 1 s ) + ( n 1 s 1 ) + + ( n 1 s r + 1 ) .

2 Proofs of theorems

In this section, we will give a proof for Theorem 1.10, which is motivated by the methods used in [37].

Proof of Theorem 1.10 Let F = { F 1 , F 2 , , F m } . We may assume (after relabeling) that for 1 i t , n F i and that for i > t , n F i . With each set F i F , we associate its characteristic vector v i = ( v i 1 , , v i n ) R n , where v i j = 1 if j F i and v i j = 0 otherwise. Let v i ¯ be the characteristic vector of F i c .

Recall that a polynomial in n variables is multilinear if its degree in each variable is at most 1. Let us restrict the domain of the polynomials we will work with to the n-cube Ω = { 0 , 1 } n R n . Since in this domain x i 2 = x i for each variable, every polynomial in our proof is multilinear.

For each F i F , define
f i ( x ) = l j L ( v i x ( | F i | l j ) ) .

Then f i ( v i ¯ ) 0 for every 1 i m and f i ( v j ¯ ) = 0 for i j , since K L = . Thus, { f i ( x ) | 1 i m } is a linearly independent family.

Let G = { G 1 , , G p } be the family of subsets of X with size at most s that contain n, which is ordered by size, that is, | G i | | G j | if i < j , where p = i = 0 s 1 ( n 1 i ) . Let u i denote the characteristic vector of G i . For i = 1 , , p , we define
g i ( x ) = j G i x j .

Since g i ( u i ) 0 for every 1 i p and g i ( u j ) = 0 for any j < i , { g i ( x ) | 1 i p } is a linearly independent family.

Let H = { H 1 , , H q } be the family of subsets of [ n ] { n } with size at most s 2 r , where q = i = 0 s 2 r ( n 1 i ) . We order the members of such that | H i | | H j | if i < j . Let w i be the characteristic vector of H i .

Let W = { n k i 1 | k i K } { n k i | k i K } . Then | W | 2 r . Set
f ( x ) = h W ( j = 1 n 1 x j h ) .
For i = 1 , , q , define
h i ( x ) = f ( x ) j H i x j .

Note that h i ( w j ) = 0 for any j > i and h i ( w i ) 0 for every 1 i q since n s + max k i , and thus { h i ( x ) | 1 i q } is a linearly independent family.

We will show that the polynomials in
{ f i ( x ) | 1 i m } { g i ( x ) | 1 i p } { h i ( x ) | 1 i q }
are linearly independent. Suppose that we have a linear combination of these polynomials that equals zero:
i = 1 m α i f i ( x ) + i = 1 p β i g i ( x ) + i = 1 q γ i h i ( x ) = 0 .
(2.1)
We will prove that the coefficients must be zero. First substitute the characteristic vector v i ¯ of F i c with n F i into equation (2.1). Since n G j , g j ( v i ¯ ) = 0 for every 1 j p . Note that we have h j ( v i ¯ ) = 0 for every 1 j q . Recall that f j ( v i ¯ ) = 0 for j i , we obtain α i f i ( v i ¯ ) = 0 . Thus, α i = 0 for every 1 i t , since f i ( v i ¯ ) 0 . It follows that
i = t + 1 m α i f i ( x ) + i = 1 p β i g i ( x ) + i = 1 q γ i h i ( x ) = 0 .
(2.2)
Then we substitute the characteristic vector v i ¯ of F i c { n } with n F i into equation (2.2). For every 1 j q and 1 k p , h j ( v i ¯ ) = 0 , g k ( v i ¯ ) = 0 . Note that f j ( v i ¯ ) = 0 for j i , we obtain α i f i ( v i ¯ ) = 0 . Since f i ( v i ¯ ) 0 , α i = 0 for every t + 1 i m . Therefore, equation (2.2) reduces to
i = 1 p β i g i ( x ) + i = 1 q γ i h i ( x ) = 0 .
(2.3)
First, we substitute the characteristic vector w i of H i with the smallest size into equation (2.3). We follow the same process to substitute the characteristic vector w i of H i with the smallest size after deleting first H i . Note that h i ( w j ) = 0 for any j > i and h i ( w i ) 0 for every 1 i q , since H i does not contain n and G i contains n, g k ( w i ) = 0 for 1 k p . Thus, we reduce (2.3) to
i = 1 p β i g i ( x ) = 0 .
(2.4)
We prove that { g i ( x ) | 1 i p } is already a linearly independent family. To complete the proof, simply note that each polynomial in { f i ( x ) | 1 i m } { g i ( x ) | 1 i p } { h i ( x ) | 1 i q } can be written as a linear combination of the multilinear polynomials of degree at most s. The space of such multilinear polynomials has dimension i = 0 s ( n i ) . It follows that
m + p + q = | F | + i = 0 s 1 ( n 1 i ) + i = 0 s 2 r ( n 1 i ) i = 0 s ( n i )
which implies
| F | ( n 1 s ) + ( n 1 s 1 ) + + ( n 1 s 2 r + 1 ) .

This completes the proof of the theorem. □

Proof of Theorem 1.11 Let n F F F . Then consider F = { F 1 , F 2 , , F m } where F i = F i { n } for 1 i m . Now, | F i | K = { k 1 , k 2 , , k r } , where k i = k i 1 . Similarly, | F i F j | L = { l 1 , l 2 , , l s } , where l i = l i 1 . Since n s + max k i , then n 1 s + max k i . Thus, it follows from Theorem 1.9 that
m ( n 1 s ) + ( n 1 s 1 ) + + ( n 1 s r + 1 ) .

 □

Declarations

Acknowledgements

This work was supported by the Dong-A University research fund.

Authors’ Affiliations

(1)
Department of Mathematics, Pearl River College, Tianjin University of Finance & Economics
(2)
Department of Mathematics, Dong-A University

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Copyright

© Liu and Hwang; licensee Springer. 2013

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