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

when we have the conditions that K\cap \mathcal{L}=\mathrm{\varnothing} and n\ge 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\cap \mathcal{L}=\mathrm{\varnothing}.

**MSC:**05D05.

## 1 Introduction

In this paper, let *X* denote the set [n]=\{1,2,\dots ,n\} and \mathcal{L}=\{{l}_{1},{l}_{2},\dots ,{l}_{s}\} be a set of *s* nonnegative integers. A family ℱ of subsets of *X* is called ℒ*-intersecting* if |E\cap F|\in \mathcal{L} for every pair of distinct subsets E,F\in \mathcal{F}. 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 \mathcal{L}=\{0,1,\dots ,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* 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 \mathcal{L}=\{0,1,\dots ,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},\dots ,{k}_{r}\}, \mathcal{L}=\{{l}_{1},{l}_{2},\dots ,{l}_{s}\} *be two sets of nonnegative integers and let* ℱ *be an* ℒ-*intersecting family of subsets of* *X* *such that* |F|\in K *for every* F\in \mathcal{F}. *If* min{k}_{i}>s-r, *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 s-r+1 and at most *s*, and \mathcal{L}=\{0,1,\dots ,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},\dots ,{k}_{r}\}, \mathcal{L}=\{{l}_{1},{l}_{2},\dots ,{l}_{s}\} *be two sets of nonnegative integers and let* ℱ *be an* ℒ-*intersecting family of subsets of* *X* *such that* |F|\in K *for every* F\in \mathcal{F}. *If* K\cap \mathcal{L}=\mathrm{\varnothing}, *then*

**Theorem 1.5** (Snevily [4])

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

**Theorem 1.6** (Hwang and Sheikh [5])

*Let* K=\{k,k+1,\dots ,k+r-1\}, \mathcal{L}=\{{l}_{1},{l}_{2},\dots ,{l}_{s}\} *be two sets of nonnegative integers and let* ℱ *be an* ℒ-*intersecting family of subsets of* *X* *such that* |F|\in K *for every* F\in \mathcal{F}. *If* K\cap \mathcal{L}=\mathrm{\varnothing} *and* k>s-r, *then*

**Theorem 1.7** (Hwang and Sheikh [5])

*Let* K=\{{k}_{1},{k}_{2},\dots ,{k}_{r}\}, \mathcal{L}=\{{l}_{1},{l}_{2},\dots ,{l}_{s}\} *be two sets of nonnegative integers and let* ℱ *be an* ℒ-*intersecting family of subsets of* *X* *such that* |F|\in K *for every* F\in \mathcal{F}. *If* {\bigcap}_{F\in \mathcal{F}}F\ne \mathrm{\varnothing} *and* min{k}_{i}>s-r, *then*

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

*Let* *K* *and* ℒ *be subsets of* \{0,1,\dots ,p-1\} *such that* K\cap \mathcal{L}=\mathrm{\varnothing}, *where* *p* *is a prime and* \mathcal{F}=\{{F}_{1},{F}_{2},\dots ,{F}_{m}\} *be a family of subsets of* [n] *such that* |{F}_{i}|(modp)\in K *for all* {F}_{i}\in \mathcal{F} *and* |{F}_{i}\cap {F}_{j}|(modp)\in \mathcal{L} *for* i\ne j. *If* n\ge s+max{k}_{i} *for every* *i*, *then* |\mathcal{F}|\le \left(\genfrac{}{}{0ex}{}{n}{s}\right)+\left(\genfrac{}{}{0ex}{}{n}{s-1}\right)+\cdots +\left(\genfrac{}{}{0ex}{}{n}{s-r+1}\right).

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},\dots ,{k}_{r}\}, \mathcal{L}=\{{l}_{1},{l}_{2},\dots ,{l}_{s}\} *be two sets of nonnegative integers and let* ℱ *be an* ℒ-*intersecting family of subsets of* *X* *such that* |F|\in K *for every* F\in \mathcal{F}. *If* n\ge s+max{k}_{i}, *then*

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

**Theorem 1.10** *Let* K=\{{k}_{1},{k}_{2},\dots ,{k}_{r}\}, \mathcal{L}=\{{l}_{1},{l}_{2},\dots ,{l}_{s}\} *be two sets of nonnegative integers and let* ℱ *be an* ℒ-*intersecting family of subsets of* *X* *such that* |F|\in K *for every* F\in \mathcal{F}. *If* K\cap \mathcal{L}=\mathrm{\varnothing} *and* n\ge s+max{k}_{i}, *then*

We note that in some cases the conditions K\cap \mathcal{L}=\mathrm{\varnothing} and n\ge 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 \mathcal{L}=\{0,1,2,4,5\}, then it is clear that K\cap \mathcal{L}=\mathrm{\varnothing} and s+max{k}_{i}=5+6<12=n, but min{k}_{i}=3<5=max{l}_{j}.

If the condition K\cap \mathcal{L}=\mathrm{\varnothing} in Theorem 1.10 is replaced by {\bigcap}_{F\in \mathcal{F}}F\ne \mathrm{\varnothing}, 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},\dots ,{k}_{r}\}, \mathcal{L}=\{{l}_{1},{l}_{2},\dots ,{l}_{s}\} *be two sets of nonnegative integers and let* ℱ *be an* ℒ-*intersecting family of subsets of* *X* *such that* |F|\in K *for every* F\in \mathcal{F}. *If* {\bigcap}_{F\in \mathcal{F}}F\ne \mathrm{\varnothing} *and* n\ge s+max{k}_{i}, *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 \mathcal{F}=\{{F}_{1},{F}_{2},\dots ,{F}_{m}\}. We may assume (after relabeling) that for 1\le i\le t, n\in {F}_{i} and that for i>t, n\notin {F}_{i}. With each set {F}_{i}\in \mathcal{F}, we associate its characteristic vector {v}_{i}=({v}_{{i}_{1}},\dots ,{v}_{{i}_{n}})\in {\mathbb{R}}^{n}, where {v}_{{i}_{j}}=1 if j\in {F}_{i} and {v}_{{i}_{j}}=0 otherwise. Let \overline{{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 \mathrm{\Omega}={\{0,1\}}^{n}\subseteq {\mathbb{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}\in \mathcal{F}, define

Then {f}_{i}(\overline{{v}_{i}})\ne 0 for every 1\le i\le m and {f}_{i}(\overline{{v}_{j}})=0 for i\ne j, since K\cap \mathcal{L}=\mathrm{\varnothing}. Thus, \{{f}_{i}(x)|1\le i\le m\} is a linearly independent family.

Let \mathcal{G}=\{{G}_{1},\dots ,{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}|\le |{G}_{j}| if i<j, where p={\sum}_{i=0}^{s-1}\left(\begin{array}{c}n-1\\ i\end{array}\right). Let {u}_{i} denote the characteristic vector of {G}_{i}. For i=1,\dots ,p, we define

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

Let \mathcal{H}=\{{H}_{1},\dots ,{H}_{q}\} be the family of subsets of [n]-\{n\} with size at most s-2r, where q={\sum}_{i=0}^{s-2r}\left(\begin{array}{c}n-1\\ i\end{array}\right). We order the members of ℋ such that |{H}_{i}|\le |{H}_{j}| if i<j. Let {w}_{i} be the characteristic vector of {H}_{i}.

Let W=\{n-{k}_{i}-1|{k}_{i}\in K\}\cup \{n-{k}_{i}|{k}_{i}\in K\}. Then |W|\le 2r. Set

For i=1,\dots ,q, define

Note that {h}_{i}({w}_{j})=0 for any j>i and {h}_{i}({w}_{i})\ne 0 for every 1\le i\le q since n\ge s+max{k}_{i}, and thus \{{h}_{i}(x)|1\le i\le q\} 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 \overline{{v}_{i}} of {F}_{i}^{c} with n\in {F}_{i} into equation (2.1). Since n\in {G}_{j}, {g}_{j}(\overline{{v}_{i}})=0 for every 1\le j\le p. Note that we have {h}_{j}(\overline{{v}_{i}})=0 for every 1\le j\le q. Recall that {f}_{j}(\overline{{v}_{i}})=0 for j\ne i, we obtain {\alpha}_{i}{f}_{i}(\overline{{v}_{i}})=0. Thus, {\alpha}_{i}=0 for every 1\le i\le t, since {f}_{i}(\overline{{v}_{i}})\ne 0. It follows that

Then we substitute the characteristic vector {\overline{{v}_{i}}}^{\star} of {F}_{i}^{c}-\{n\} with n\notin {F}_{i} into equation (2.2). For every 1\le j\le q and 1\le k\le p, {h}_{j}({\overline{{v}_{i}}}^{\star})=0, {g}_{k}({\overline{{v}_{i}}}^{\star})=0. Note that {f}_{j}({\overline{{v}_{i}}}^{\star})=0 for j\ne i, we obtain {\alpha}_{i}{f}_{i}({\overline{{v}_{i}}}^{\star})=0. Since {f}_{i}({\overline{{v}_{i}}}^{\star})\ne 0, {\alpha}_{i}=0 for every t+1\le i\le m. Therefore, equation (2.2) reduces to

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})\ne 0 for every 1\le i\le q, since {H}_{i} does not contain *n* and {G}_{i} contains *n*, {g}_{k}({w}_{i})=0 for 1\le k\le p. Thus, we reduce (2.3) to

We prove that \{{g}_{i}(x)|1\le i\le p\} is already a linearly independent family. To complete the proof, simply note that each polynomial in \{{f}_{i}(x)|1\le i\le m\}\cup \{{g}_{i}(x)|1\le i\le p\}\cup \{{h}_{i}(x)|1\le i\le 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 {\sum}_{i=0}^{s}\left(\begin{array}{c}n\\ i\end{array}\right). It follows that

which implies

This completes the proof of the theorem. □

*Proof of Theorem 1.11* Let n\in {\bigcap}_{F\in \mathcal{F}}F. Then consider {\mathcal{F}}^{\ast}=\{{F}_{1}^{\ast},{F}_{2}^{\ast},\dots ,{F}_{m}^{\ast}\} where {F}_{i}^{\ast}={F}_{i}\setminus \{n\} for 1\le i\le m. Now, |{F}_{i}^{\ast}|\in {K}^{\ast}=\{{k}_{1}^{\ast},{k}_{2}^{\ast},\dots ,{k}_{r}^{\ast}\}, where {k}_{i}^{\ast}={k}_{i}-1. Similarly, |{F}_{i}^{\ast}\cap {F}_{j}^{\ast}|\in {\mathcal{L}}^{\ast}=\{{l}_{1}^{\ast},{l}_{2}^{\ast},\dots ,{l}_{s}^{\ast}\}, where {l}_{i}^{\ast}={l}_{i}-1. Since n\ge s+max{k}_{i}, then n-1\ge s+max{k}_{i}^{\ast}. Thus, it follows from Theorem 1.9 that

□

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## Acknowledgements

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

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### Authors’ contributions

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