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

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

$$|\mathcal{F}|\le \left(\begin{array}{c}n-1\\ s\end{array}\right)+\left(\begin{array}{c}n-1\\ s-1\end{array}\right)+\cdots +\left(\begin{array}{c}n-1\\ s-2r+1\end{array}\right)$$

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

$$|\mathcal{F}|\le \left(\begin{array}{c}n\\ s\end{array}\right).$$

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

$$|\mathcal{F}|\le \left(\begin{array}{c}n\\ s\end{array}\right)+\left(\begin{array}{c}n\\ s-1\end{array}\right)+\cdots +\left(\begin{array}{c}n\\ 0\end{array}\right).$$

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

$$|\mathcal{F}|\le \left(\begin{array}{c}n\\ s\end{array}\right)+\left(\begin{array}{c}n\\ s-1\end{array}\right)+\cdots +\left(\begin{array}{c}n\\ s-r+1\end{array}\right).$$

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

$$|\mathcal{F}|\le \left(\begin{array}{c}n-1\\ s\end{array}\right)+\left(\begin{array}{c}n-1\\ s-1\end{array}\right)+\cdots +\left(\begin{array}{c}n-1\\ 0\end{array}\right).$$

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

$$|\mathcal{F}|\le \left(\begin{array}{c}n-1\\ s\end{array}\right)+\left(\begin{array}{c}n-1\\ s-1\end{array}\right)+\cdots +\left(\begin{array}{c}n-1\\ s-2r+1\end{array}\right).$$

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

$$|\mathcal{F}|\le \left(\begin{array}{c}n-1\\ s\end{array}\right)+\left(\begin{array}{c}n-1\\ s-1\end{array}\right)+\cdots +\left(\begin{array}{c}n-1\\ s-r\end{array}\right).$$

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

$$|\mathcal{F}|\le \left(\begin{array}{c}n-1\\ s\end{array}\right)+\left(\begin{array}{c}n-1\\ s-1\end{array}\right)+\cdots +\left(\begin{array}{c}n-1\\ s-r\end{array}\right).$$

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

$$|\mathcal{F}|\le \left(\begin{array}{c}n\\ s\end{array}\right)+\left(\begin{array}{c}n\\ s-1\end{array}\right)+\cdots +\left(\begin{array}{c}n\\ s-r+1\end{array}\right).$$

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

$$|\mathcal{F}|\le \left(\begin{array}{c}n-1\\ s\end{array}\right)+\left(\begin{array}{c}n-1\\ s-1\end{array}\right)+\cdots +\left(\begin{array}{c}n-1\\ s-2r+1\end{array}\right).$$

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

$$|\mathcal{F}|\le \left(\begin{array}{c}n-1\\ s\end{array}\right)+\left(\begin{array}{c}n-1\\ s-1\end{array}\right)+\cdots +\left(\begin{array}{c}n-1\\ s-r+1\end{array}\right).$$

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

$$f_i(x)=\prod _{l_j\in \mathcal{L}}(v_i\cdot x-(|F_i|-l_j)).$$

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

$$g_i(x)=\prod _{j\in G_i}{x}_j.$$

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

$$f(x)=\prod _{h\in W}(\sum _{j=1}^{n-1}{x}_j-h).$$

For $i=1,\dots ,q$, define

$$h_i(x)=f(x)\prod _{j\in H_i}{x}_j.$$

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

$$\{f_i(x)|1\le i\le m\}\cup \{g_i(x)|1\le i\le p\}\cup \{h_i(x)|1\le i\le q\}$$

are linearly independent. Suppose that we have a linear combination of these polynomials that equals zero:

$$\sum _{i=1}^m{\alpha }_i{f}_i(x)+\sum _{i=1}^p{\beta }_i{g}_i(x)+\sum _{i=1}^q{\gamma }_i{h}_i(x)=0.$$

(2.1)

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

$$\sum _{i=t+1}^m{\alpha }_i{f}_i(x)+\sum _{i=1}^p{\beta }_i{g}_i(x)+\sum _{i=1}^q{\gamma }_i{h}_i(x)=0.$$

(2.2)

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

$$\sum _{i=1}^p{\beta }_i{g}_i(x)+\sum _{i=1}^q{\gamma }_i{h}_i(x)=0.$$

(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 $h_i(w_i)\ne 0$ 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

$$\sum _{i=1}^p{\beta }_i{g}_i(x)=0.$$

(2.4)

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

$$m+p+q=|\mathcal{F}|+\sum _{i=0}^{s-1}\left(\begin{array}{c}n-1\\ i\end{array}\right)+\sum _{i=0}^{s-2r}\left(\begin{array}{c}n-1\\ i\end{array}\right)\le \sum _{i=0}^s\left(\begin{array}{c}n\\ i\end{array}\right)$$

which implies

$$|\mathcal{F}|\le \left(\begin{array}{c}n-1\\ s\end{array}\right)+\left(\begin{array}{c}n-1\\ s-1\end{array}\right)+\cdots +\left(\begin{array}{c}n-1\\ s-2r+1\end{array}\right).$$

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

$$m\le \left(\begin{array}{c}n-1\\ s\end{array}\right)+\left(\begin{array}{c}n-1\\ s-1\end{array}\right)+\cdots +\left(\begin{array}{c}n-1\\ s-r+1\end{array}\right).$$

 □