Stability versions of Erdős–Ko–Rado type theorems via isoperimetry

Abstract

Erdős-Ko-Rado (EKR) type theorems yield upper bounds on the sizes of families of sets, subject to various intersection requirements on the sets in the family. Stability versions of such theorems assert that if the size of a family is close to the maximum possible size, then the family itself must be close (in some appropriate sense) to a maximum-sized family.In this paper, we present an approach to obtaining stability versions of EKR-type theorems, via isoperimetric inequalities for subsets of the hypercube. Our approach is rather general, and allows the leveraging of a wide variety of exact EKR-type results into strong stability versions of these results, without going into the proofs of the original results.We use this approach to obtain tight stability versions of the EKR theorem itself and of the Ahlswede-Khachatrian theorem on t-intersecting families of k-element subsets of {1, 2, . . . , n} (for k < n t+1 ), and to show that, somewhat surprisingly, all these results hold when the 'intersection' requirement is replaced by a much weaker requirement.Other examples include stability versions of Frankl's recent result on the Erdős matching conjecture, the Ellis-Filmus-Friedgut proof of the Simonovits-Sós conjecture, and various EKR-type results on r-wise (cross)-t-intersecting families.The two conditions above hold in a wide variety of settings, including:1. The EKR theorem itself (where the extremal example is a dictatorship), 2. Wilson's theorem on t-intersecting families (where the extremal example is a t-umvirate), The Simonovits-Sós conjecture on triangle-intersecting families of graphs, proved recently by Ellis, Filmus and Friedgut [19] (where the extremal example is a specific type of 3umvirate, namely all graphs containing a fixed triangle), 4. The Erdős matching conjecture on the maximal size of a family without s pairwise disjoint sets, proved recently by Frankl [27] for n > (2s + 1)k − s (where the extremal example is the dual of an (s − 1)-umvirate),5. All EKR-type results on r-wise (cross)-t-intersecting families known to us, for which the extremal example is a t-umvirate. (See e.g. [28,39,55,56] and the references therein.)

On set systems without a simplex-cluster and the junta method

“…to showing that a (d + 1)-wise intersecting family whose size is close to n−1 k−1 is close to a star. This was proved by Ellis, Keller, and the author [5].…”

“…Proof of Proposition 4.1. We shall need the following stability result for Frankl's Theorem that was essentially proved by Ellis, Keller, and the author [5]. We shall use the following corollary of their work stated by Keller and the author in [18].…”

On the chromatic number of a random subgraph of the Kneser graph

“…The best known results in this direction were obtained in [15] and, more recently, in [19]. See also [11,17,18] for related stability results.…”

Regular intersecting families

“…, which is the same as the degree of, say, 2k − 1, equal by (9) and (10) to (10), we only need to show that there exists Q of desired size and without A i , in which any two degrees differ by at most 1.…”