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A new Alon-Babai-Suzuki-type inequality on set systems
Journal of Inequalities and Applications volume 2013, Article number: 171 (2013)
Abstract
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 .
MSC:05D05.
1 Introduction
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 [1])
If ℱ is a k-uniform ℒ-intersecting family of subsets of X, then
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 [2])
If ℱ is an ℒ-intersecting family of subsets of X, then
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 [3])
Let , be two sets of nonnegative integers and let ℱ be an ℒ-intersecting family of subsets of X such that for every . If , then
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 1.4 Let , be two sets of nonnegative integers and let ℱ be an ℒ-intersecting family of subsets of X such that for every . If , then
Theorem 1.5 (Snevily [4])
Let , be two sets of nonnegative integers and let ℱ be an ℒ-intersecting family of subsets of X such that for every . If , then
Theorem 1.6 (Hwang and Sheikh [5])
Let , be two sets of nonnegative integers and let ℱ be an ℒ-intersecting family of subsets of X such that for every . If and , then
Theorem 1.7 (Hwang and Sheikh [5])
Let , be two sets of nonnegative integers and let ℱ be an ℒ-intersecting family of subsets of X such that for every . If and , then
Conjecture 1.8 (Alon, Babai, and Suzuki [3])
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. [6])
Let , be two sets of nonnegative integers and let ℱ be an ℒ-intersecting family of subsets of X such that for every . If , then
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.
Theorem 1.10 Let , be two sets of nonnegative integers and let ℱ be an ℒ-intersecting family of subsets of X such that for every . If and , then
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.
Theorem 1.11 Let , be two sets of nonnegative integers and let ℱ be an ℒ-intersecting family of subsets of X such that for every . If and , then
2 Proofs of theorems
In this section, we will give a proof for Theorem 1.10, which is motivated by the methods used in [3–7].
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.
For each , define
Then for every and for , since . Thus, is a linearly independent family.
Let be the family of subsets of X with size at most s that contain n, which is ordered by size, that is, if , where . Let denote the characteristic vector of . For , we define
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 .
Let . Then . Set
For , define
Note that for any and for every since , and thus is a linearly independent family.
We will show that the polynomials in
are linearly independent. Suppose that we have a linear combination of these polynomials that equals zero:
We will prove that the coefficients must be zero. First substitute the characteristic vector of with into equation (2.1). Since , for every . Note that we have for every . Recall that for , we obtain . Thus, for every , since . It follows that
Then we substitute the characteristic vector of with into equation (2.2). For every and , , . Note that for , we obtain . Since , for every . Therefore, equation (2.2) reduces to
First, we substitute the characteristic vector of with the smallest size into equation (2.3). We follow the same process to substitute the characteristic vector of with the smallest size after deleting first . Note that for any and for every , since does not contain n and contains n, for . Thus, we reduce (2.3) to
We prove that is already a linearly independent family. To complete the proof, simply note that each polynomial in can be written as a linear combination of the multilinear polynomials of degree at most s. The space of such multilinear polynomials has dimension . It follows that
which implies
This completes the proof of the theorem. □
Proof of Theorem 1.11 Let . Then consider where for . Now, , where . Similarly, , where . Since , then . Thus, it follows from Theorem 1.9 that
□
References
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Alon N, Babai L, Suzuki H: Multilinear polynomials and Frankl-Ray-Chaudhuri-Wilson-Type intersection theorems. J. Comb. Theory, Ser. A 1991, 58: 165–180. 10.1016/0097-3165(91)90058-O
Snevily HS: A generalization of the Ray-Chaudhuri-Wilson theorem. J. Comb. Des. 1995, 3: 349–352. 10.1002/jcd.3180030505
Hwang K-W, Sheikh NN: Intersection families and Snevily’s conjecture. Eur. J. Comb. 2007, 28: 843–847. 10.1016/j.ejc.2005.11.002
Hwang K-W, Kim T, Jang LC, Kim P, Sohn G: Alon-Babai-Suzuki’s conjecture related to binary codes in nonmodular version. J. Inequal. Appl. 2010., 2010: Article ID 546015. doi:10.1155/2010/546015
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Acknowledgements
This work was supported by the Dong-A University research fund.
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KWH made the first polynomial and proved the first polynomials are linear independent. RXJL made the other polynomials and proved that the other polynomials are linearly independent. All authors read and approved the final manuscript.
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Liu, R.X., Hwang, KW. A new Alon-Babai-Suzuki-type inequality on set systems. J Inequal Appl 2013, 171 (2013). https://doi.org/10.1186/1029-242X-2013-171
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DOI: https://doi.org/10.1186/1029-242X-2013-171