Regular intersecting families

Abstract

We call a family of sets intersecting, if any two sets in the family intersect. In this paper we investigate intersecting families F of k-element subsets of [n] := {1, . . . , n}, such that every element of [n] lies in the same (or approximately the same) number of members of F . In particular, we show that we can guarantee |F | = o( n−1 k−1 ) if and only if k = o(n).

Refuting conjectures in extremal combinatorics via linear programming

“…A Hoffman-type eigenvalue bound on regular set systems. We say that a family F ⊂ 2 [n] is s-subset-regular if every set of size s lies in the same number of elements of F. Ihringer and Kupavskii [25] proved the following Hoffman-type eigenvalue upper bound on such regular families: .…”

“…They proved [25] that equality in Theorem 3.17 is achieved with (n, k, s) = (7, 3, 1) and (9,4,1). They asked whether there are other values of the parameters with n ≥ 2k + 1 for which Theorem 3.17 is tight.…”

Degree versions of theorems on intersecting families via stability

“…Erdős-Ko-Rado theorem spurred the development of extremal combinatorics, and by now there are numerous variations and extensions of Theorem 1.1 and the Hilton-Milner theorem (cf. [2,4,5,6,7,8,15,16,23,28,29,30,32,34,35,36,37,38,39,40] to name a few recent ones). We refer the reader to a recent survey by Frankl and Tokushige [22].…”

“…The bound in the theorem is tight because of the trivial intersecting family, and the condition n > 2k is necessary: the authors of [27] provide an example of such family for n = 2k which has larger minimum degree. In fact, for most values of k there are regular intersecting families in [2k] k of maximum possible size 2k−1 k−1 (see [28]). In the follow-up paper, Frankl, Han, Huang, and Zhao [14] proved the following theorem.…”

On regular 3-wise intersecting families

“…J.K. was supported by NSF Grant DMS1501962. 1 need not all be of the same size; for related work on uniform intersecting families, see the paper of Ellis, Kalai and the third author [4] addressing the symmetric case, and the results of Ihringer and Kupavskii [10] addressing the regular case.…”