A new Alon-Babai-Suzuki-type inequality on set systems
© Liu and Hwang; licensee Springer. 2013
Received: 1 February 2013
Accepted: 2 April 2013
Published: 15 April 2013
Let and be two sets of nonnegative integers. Let ℱ be a family of subsets of such that for every and for every pair of distinct subsets . We prove that
when we have the conditions that and . 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 .
KeywordsAlon-Babai-Suzuki inequalities Frankl-Ray-Chaudhuri-Wilson theorems multilinear polynomials
In this paper, let X denote the set and be a set of s nonnegative integers. A family ℱ of subsets of X is called ℒ-intersecting if for every pair of distinct subsets . A family ℱ 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 )
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 . A nonuniform version of Theorem 1.1 was proved by Frankl and Wilson in 1981.
Theorem 1.2 (Frankl and Wilson )
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 .
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 )
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 and at most s, and .
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 1.5 (Snevily )
Theorem 1.6 (Hwang and Sheikh )
Theorem 1.7 (Hwang and Sheikh )
Conjecture 1.8 (Alon, Babai, and Suzuki )
Let K and ℒ be subsets of such that , where p is a prime and be a family of subsets of such that for all and for . If for every i, then .
In the following paper, they prove the above conjecture in the nonmodular.
Theorem 1.9 (Hwang et al. )
In this paper, we will prove the following Alon-Babai-Suzuki-type inequality which gives an improvement for Theorem 1.9 with and for Alon, Babai and Suzuki’s conjecture in the nonmodular.
We note that in some cases the conditions and in Theorem 1.10 holds, but Snevily’s condition in Theorem 1.5 does not. For instance, if , and , then it is clear that and , but .
If the condition in Theorem 1.10 is replaced by , we have the following result which gives a better bound than Theorem 1.10 and Theorem 1.7.
2 Proofs of theorems
Proof of Theorem 1.10 Let . We may assume (after relabeling) that for , and that for , . With each set , we associate its characteristic vector , where if and otherwise. Let be the characteristic vector of .
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 . Since in this domain for each variable, every polynomial in our proof is multilinear.
Then for every and for , since . Thus, is a linearly independent family.
Since for every and for any , is a linearly independent family.
Let be the family of subsets of with size at most , where . We order the members of ℋ such that if . Let be the characteristic vector of .
Note that for any and for every since , and thus is a linearly independent family.
This completes the proof of the theorem. □
This work was supported by the Dong-A University research fund.
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