On the d-cluster generalization of Erdős-Ko-Rado

Abstract

If $2\le d\le k$ and $n\ge dk/(d-1)$, a d-cluster is defined to be a collection of d elements of $\left(\begin{array}{c}[n]\\ k\end{array}\right)$ with empty intersection and union of size no more than 2k. Mubayi [14] conjectured that the largest size of a d-cluster-free family $\mathcal{F}\subset \left(\begin{array}{c}[n]\\ k\end{array}\right)$ is $\left(\begin{array}{c}n-1\\ k-1\end{array}\right)$, with equality holding only for a maximum-sized star. Here, we resolve Mubayi's conjecture and prove a slightly stronger result, thus completing a new generalization of the Erdős-Ko-Rado Theorem.

Introduction

For any $m,n\in \mathbb{Z}$ with $m<n$, we define $[m,n]:=\{m,m+1,\dots ,n\}$, and $[n]:=[1,n]$, and we use $\left(\begin{array}{c}X\\ k\end{array}\right)$ to refer to the set of k-element subsets of a set X. Furthermore, if every element of a family $\mathcal{F}\subset \left(\begin{array}{c}X\\ k\end{array}\right)$ contains some $x\in X$, we say that $\mathcal{F}$ is a star (centered at x). We note that the maximum size of a star in $\left(\begin{array}{c}[n]\\ k\end{array}\right)$ is $\left(\begin{array}{c}n-1\\ k-1\end{array}\right)$ and recall the classical Erdős-Ko-Rado (EKR) theorem, which gives an upper bound on the size of “pairwise intersecting” set systems.

Theorem 1 [5]

Let $n\ge 2k$ and suppose that $\mathcal{F}\subset \left(\begin{array}{c}[n]\\ k\end{array}\right)$ has the property that $A\cap B\ne \varnothing$ for all $A,B\in \mathcal{F}$. Then

$$|\mathcal{F}|\le \left(\begin{array}{c}n-1\\ k-1\end{array}\right),$$

where, excluding the case $n=2k$, equality is achieved only when $\mathcal{F}$ is a maximum-sized star.

Theorem 1 is one of the fundamental results in extremal combinatorics, and has spawned numerous generalizations and conjectures. In particular, it has generated a whole area of “intersection problems” for set systems, in which we consider the maximum size of a family where we have forbidden a certain class of subfamilies, defined according to some intersection and union constraints. One example of such a class which is directly related to the original EKR theorem is the notion of a d-cluster, introduced by Mubayi in [14].

Definition 1

Let $2\le d\le k$ with $n\ge dk/(d-1)$ and suppose that $\mathcal{F}\subset \left(\begin{array}{c}[n]\\ k\end{array}\right)$. Then, if we have $\mathcal{B}=\{B_1,\dots ,B_d\}\subset \left(\begin{array}{c}[n]\\ k\end{array}\right)$ such that $|B_1\cup \dots \cup B_d|\le 2k$ and $B_1\cap \dots \cap B_d=\varnothing$, we say that $\mathcal{B}$ is a d-cluster. Furthermore, if $\mathcal{F}$ contains no such $\mathcal{B}$, we say that $\mathcal{F}$ is d-cluster-free.

Note here that in the case of $d=2$, the union condition holds automatically. Thus, another way to state Theorem 1 would be to say that 2-cluster-free families can have size no greater than $\left(\begin{array}{c}n-1\\ k-1\end{array}\right)$. In [14], Mubayi showed that 3-cluster-free families must also obey this bound, and conjectured that the same would hold for $d\ge 4$. The primary goal of this paper is to resolve Mubayi's conjecture.

Theorem 2

Let $2\le d\le k$ and $n\ge dk/(d-1)$. Furthermore, suppose that $\mathcal{F}\subset \left(\begin{array}{c}[n]\\ k\end{array}\right)$ is d-cluster-free. Then

$$|\mathcal{F}|\le \left(\begin{array}{c}n-1\\ k-1\end{array}\right),$$

and, excepting the case when both $d=2$ and $n=2k$, equality implies $\mathcal{F}$ is a maximum-sized star.

We note that the conditions $d\le k$ and $n\ge dk/(d-1)$ are in fact necessary; if we let $d=k+1$ then the problem is equivalent to one of the long-open hypergraph Turán problems, and a different bound applies. Furthermore, if $n<dk/(d-1)$, then any collection of d sets cannot have empty intersection. A number of previous results on this problem have been shown; Katona first proposed the problem as a version of the $d=3$ case, and Frankl and Füredi in [8] obtained the desired bound for $d=3$ and $n\ge k^2+3k$. Mubayi, in addition to resolving the $d=3$ case and proposing the more general problem in [14], showed in [15] that the theorem holds when $d=4$ and n is sufficiently large. Later, Mubayi and Ramadurai [16] and independently Özkahya and Füredi [11] showed that it also holds when $d>4$ and n is sufficiently large. In [12], Keevash and Mubayi solved another case of this problem, namely where both $k/n$ and $n/2-k$ are bounded away from zero. The case of $n<2k$ (where, again, the union condition holds automatically) was resolved by Frankl in [6] (where the bound was established) and in [7] (where equality was characterized).

We note finally that the problem of d-clusters is closely related to several other old problems in extremal set theory, in particular the simplex and special simplex conjectures of Chvátal [3] and Frankl and Füredi [9], and more generally the so-called Turán problems for expansion. For a detailed historical account of these kinds of problems, as well as some exciting new results, we direct the interested reader to [13]. We also note that most remaining cases of Chvátal's conjecture were shown in a recent paper of the author [4], in which an alternative (but related) proof of Theorem 2 for a more limited range of parameters is also given. However, the present manuscript, and the techniques described here, appeared first.

Despite the success in tackling the problem of clusters for large values of n, techniques that could obtain the desired bound for $d\ge 4$ and all $n\ge 2k$ have remained largely elusive. In this paper, we bridge the gap by giving a slightly more general result that is valid for all $d\ge 2$ and $n\ge 2k$, which in combination with the known results on $n<2k$ will give us Theorem 2. Our more general theorem will give us more powerful inductive tools that we will leverage in our proof, and is based on a method of induction introduced in [15].

Theorem 3

Let $2\le d\le k\le n/2$. Furthermore, suppose that $\mathcal{F}\subset \left(\begin{array}{c}[n]\\ k\end{array}\right)$, and that ${\mathcal{F}}^{⁎}\subset \mathcal{F}$ has the property that any d-cluster in $\mathcal{F}$ is contained entirely in ${\mathcal{F}}^{⁎}$. Then,

$$|{\mathcal{F}}^{⁎}|+\frac{n}{k}|\mathcal{F}\setminus {\mathcal{F}}^{⁎}|\le \left(\begin{array}{c}n\\ k\end{array}\right).$$

Furthermore, excepting the case where both $d=2$ and $n=2k$, equality implies one of the following:

• ${\mathcal{F}}^{⁎}=\varnothing$ and $\mathcal{F}$ is a maximum-sized star,

• $\mathcal{F}={\mathcal{F}}^{⁎}=\left(\begin{array}{c}[n]\\ k\end{array}\right)$.

Note that Theorem 3 reduces to Theorem 2 for all $n\ge 2k$ if we set ${\mathcal{F}}^{⁎}=\varnothing$. The $d=2$ case of Theorem 3 is itself an interesting strengthening of Erdős-Ko-Rado, and has actually appeared before in the literature. To our knowledge, it was shown first by Borg in [1] using shadow techniques, and by Borg and Leader in [2] using cycle methods, and can also be seen as a consequence of a more general result in [10] (Theorem 9 with $c=\frac{n-k}{k}$). These results are all related to cross-intersecting families; that is, families $\mathcal{F},\mathcal{G}\subset \left(\begin{array}{c}[n]\\ k\end{array}\right)$ such that for all $A\in \mathcal{F}$ and $B\in \mathcal{G}$ we have $A\cap B\ne \varnothing$. In fact, the condition required for Theorem 3 to hold (in the $d=2$ case) is equivalent to requiring that $\mathcal{F}$ and $\mathcal{F}\setminus {\mathcal{F}}^{⁎}$ are cross-intersecting. The connection between Theorem 3 and the notion of cross-intersecting families is an interesting one and perhaps deserves further attention.