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

- Rudy XJ Liu
^{1}and - Kyung-Won Hwang
^{2}Email author

**2013**:171

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

© Liu and Hwang; licensee Springer. 2013

**Received: **1 February 2013

**Accepted: **2 April 2013

**Published: **15 April 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.

## Keywords

## 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.

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.

*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 ${H}_{i}$.

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.

*n*and ${G}_{i}$ contains

*n*, ${g}_{k}({w}_{i})=0$ for $1\le k\le p$. Thus, we reduce (2.3) to

*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

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

□

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

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