On Matchings in Hypergraphs

Abstract

We show that if the largest matching in a $k$-uniform hypergraph $G$ on $n$ vertices has precisely $s$ edges, and $n>2k^2s/\log k$, then $H$ has at most $\binom n k - \binom {n-s} k $ edges and this upper bound is achieved only for hypergraphs in which the set of edges consists of all $k$-subsets which intersect a given set of $s$ vertices.

Two problems on matchings in set families – In the footsteps of Erdős and Kleitman

“…[5], [23] and [9]). Improving earlier results of [4], [2], [17] and [15], the first author [8] proved…”

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

“…For arbitrary k, Erdős [4] proved the conjecture for n ≥ n 0 (k, s); Bollobás, Daykin and Erdős [2] proved the conjecture for n > 2k 3 (s − 1). Huang, Loh and Sudakov [16] improved it to n ≥ 3k 2 s; Frankl, Luczak, and Mieczkowska [11] further improved to n ≥ 2k 2 s/ log k. Recently Frankl [9] proved the conjecture for n ≥ (2s − 1)k − s + 1.…”

On the bandwidth of the Kneser graph

“…Erdős [7] showed that there exist n 1 (r, p) such that for all n ≥ n 1 (r, p) the maximum size of a family of r-subsets of [n] not containing a matching of p + 1 edges is n r − n−p r . There has subsequently been a lot of work on determining the smallest n 1 (r, p) for which the statement holds (see [3,10,11,14] for instance). The best result among these is due to Frankl [9].…”