Pseudo sunflowers

Abstract

A collection $A_0,A_1,\dots ,A_t$ of finite sets is called a pseudo sunflower of size $t+1$ and with center $C$ if $C$ is a proper subset of $A_0$ and the sets $A_0\setminus C,\dots ,A_t\setminus C$ are pairwise disjoint. The main result shows that every collection of more than $t^k$ distinct $k$-element sets contains a pseudo sunflower of size $t+1$. The bound is best possible.

Introduction

Let $k$ and $t$ be positive integers. A collection of distinct $k$-element sets or $k$-sets for short, is called a $k$-uniform family or for short a $k$-graph.

A sunflower (of size $t+1$ and with center $C$ ) is a collection of $t+1$ distinct sets $A_0,\dots ,A_t$ such that $A_i\cap A_j=C$ for all $0\le i<j\le t$.

Let us recall a simple but very important result due to Erdős and Rado. We need a definition. Let $s(k,t)$ denote the maximum of $\left|\mathcal{F}\right|$ where $\mathcal{F}$ is a $k$-graph not containing any sunflower of size $t+1$. Note that $s(k,1)=1$ and $s(1,t)=t$ are trivial.

Erdős–Rado Sunflower Lemma ([1]).

$$s(k,t)\le k!t^k.$$

The proof is by simple induction but this result played an essential role in the proof of many combinatorial results (cf. e.g. [2]) and also in the seminal paper of Razborov [3] concerning complexity theory. Most applications are based on the following simple fact.

Deza’s Lemma (cf. [2]). Suppose that $A_0,\dots ,A_t$ form a sunflower with center $C$, and $B$ is a set of size at most $t$. Then

$$B\cap A_i=B\cap Choldsforsomei,0\le i\le t.$$

The term sunflower was coined in [4]. Abbott et al. [5] determined $s(2,t)$. For $t=2\ell$, even one has $s(2,2\ell )=2\ell (2\ell +1)$ the lower bound coming from the vertex disjoint union of two complete graphs on $2\ell +1$ vertices.

The inequality $s(k+\ell ,t)\ge s(k,t)s(\ell ,t)$ is easy to prove. It implies that either

$$s(k,t)<c{\left(t\right)}^k\text{ for some }c\left(t\right)\text{ independent of }k\text{ or }s{(k,t)}^{1/k}\to \infty .$$

Mentioning (1.3) was a constant subject in Erdős’s many papers including a list of his favorite problems [6].

Even though that there was some major progress throughout the years (cf. [7], [8]), (1.3) is still open even for $t=2$.

Let us recall that a family $\mathcal{F}$ of sets is called $r$-intersecting if $|F\cap F^{\prime }|\ge r$ for all $F,F^{\prime }\in \mathcal{F}$. The author has been using the Erdős–Rado Sunflower Lemma since a long time ([9], [10], etc.).

However it turns out that in many problems concerning $r$-intersecting families $B\cap A_i\subset B\cap C$ would be sufficient instead of (1.2). This motivates the following definition.

Definition 1.1

The sets $A_0,\dots ,A_t$ are said to form a pseudo sunflower of size $t+1$ and center $C$ if $C⫋A_0$ and the sets $A_0\setminus C,A_1\setminus C,\dots ,A_t\setminus C$ are pairwise disjoint.

We should point out that for any set $D$ with $C\subset D⫋A_0$ the sets $A_0,\dots ,A_t$ form a pseudo sunflower with center $D$ as well.

Instead of Deza’s Lemma we have the following.

Lemma 1.2

Suppose that $A_0,\dots ,A_t$ form a pseudo sunflower with center $C$ and $B$ is a set of size at most $t$. Then

$$B\cap A_i\subset B\cap C\text{ holds for some }i,0\le i\le t.$$

Proof

Since $\left|B\right|\le t$ and $A_0\setminus C,\dots ,A_t\setminus C$ are $t+1$ pairwise disjoint sets, $B\cap (A_i\setminus C)=0̸$ must hold for some $i$, $0\le i\le t$. This is equivalent to (1.4). □

Definition 1.3

Let $p(k,t)$ denote the maximum of $\left|\mathcal{F}\right|$ over all $k$-graphs $\mathcal{F}$ not containing a pseudo sunflower of size $t+1$.

Example 1.4

Let $X_1,\dots ,X_k$ be pairwise disjoint $t$-sets and let

$$\mathcal{K}=\mathcal{K}(X_1,\dots ,X_t)=\left\{E:\left|E\right|=k,|E\cap X_j|=1\text{ for }1\le j\le k\right\}.$$

Clearly, $\left|\mathcal{K}\right|=t^k$.

Claim 1.5

$\mathcal{K}$ does not contain a pseudo sunflower of size $t+1$.

Proof

Suppose for contradiction that $E_0,\dots ,E_t\in \mathcal{K}$ form a pseudo sunflower with center $C$. Then $C⫋E_0$ implies $\left|C\right|<k$. Thus we may choose $j$ to satisfy $C\cap X_j=0̸$. Then $|(E_i\setminus C)\cap X_j|=1$ for $0\le i\le t$. Since $\left|X_j\right|=t$, they cannot be pairwise disjoint. □

Rather surprisingly and in great contrast to sunflowers we can prove that $p(k,t)=t^k$ and Example 1.4 is the essentially unique $k$-graph attaining equality.

Let us state our main result in a sharper form.

Theorem 1.6

Let $t,k,r$ be positive integers and let $\mathcal{F}$ be a $k$-graph not containing any pseudo sunflower of size $t+1$. Then

$$\left|\mathcal{F}\right|\le t^k.$$

Moreover, the inequality is strict unless $\mathcal{F}$ is isomorphic to $\mathcal{K}$. If in addition $\mathcal{F}$ is $r$-intersecting, then

$$\left|\mathcal{F}\right|\le t^k-\sum _{0\le j<r}\left(\genfrac{}{}{0ex}{}{k}{j}\right){(t-1)}^{k-j}.$$

For the proof we use the following standard notation. Let $\mathcal{F}$ be a family of sets and $D,E$ subsets with $D\subset E$. Set

$$\mathcal{F}(D,E)=\{F\setminus D:F\in \mathcal{F},F\cap E=D\}.$$

Note that for $E$ fixed

$$\left|\mathcal{F}\right|=\sum _{D\subset E}\left|\mathcal{F}(D,E)\right|$$

and if $E\in \mathcal{F}$ and $\mathcal{F}$ is $r$-intersecting, then

$$\left|\mathcal{F}\right|=\sum _{D\subset E,\left|D\right|\ge r}\left|\mathcal{F}(D,E)\right|.$$

In Section 2 we provide the proof of Theorem 1.6. In Section 3 some applications are given.