Degree versions of the Erdős–Ko–Rado theorem and Erdős hypergraph matching conjecture

Abstract

We use an algebraic method to prove a degree version of the celebrated Erdős-Ko-Rado theorem: given n > 2k, every intersecting k-uniform hypergraph H on n vertices contains a vertex that lies on at most n−2 k−2 edges. This result can be viewed as a special case of the degree version of a wellknown conjecture of Erdős on hypergraph matchings. Improving the work of Bollobás, Daykin, and Erdős from 1976, we show that given integers n, k, s with n ≥ 3k 2 s, every k-uniform hypergraph H on n vertices with minimum vertex degree greater than n−1 k−1 − n−s k−1 contains s disjoint edges.

Degree versions of theorems on intersecting families via stability

“…This improved the result of Bollobás, Daykin, and Erdős [3], who arrived at the same conclusion for n ≥ 2k 3 s. The authors of [27] conjectured that the same should hold for any n > k(s + 1). Note that the family A k (k, s) does not appear in the degree version since its minimum t-degree is 0 for n ≥ k(s + 1) and t ≥ 1.…”

“…We denote by δ(F ) and ∆(F ) the minimum degree and maximum degree of an element in F . Recently, Huang and Zhao [27] gave an elegant proof of the following theorem using a linear-algebraic approach: Theorem 1.2 ([27]). Let n > 2k > 0.…”

“…Then any intersecting family F ⊂ [n] k satisfies δ(F ) ≤ n−2 k−2 . 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]).…”

“…Several questions and problems in this context were asked in [27] and [14], as well as in personal communication with Hao Huang and his presentation on the Recent Advances in Extremal Combinatorics Workshop at TSIMF, Sanya, (May 2017). Some of them are as follows:…”

A degree version of the Hilton–Milner theorem

“…Huang and Zhao [15] recently proved a minimum degree version of the EKR theorem, which states that, if n > 2k and F is a k-uniform intersecting family on [n], then δ(F ) ≤ n−2 k−2 , and the equality holds only if F is a full star. This result implies the EKR theorem immediately: given a k-uniform intersecting family F , by recursively deleting elements with the smallest degree until 2k elements are left, we derive that…”