A Note on Perfect Matchings in Uniform Hypergraphs with Large Minimum Collective Degree

For an integer k ≥ 2 and a k-uniform hypergraph H, let δk−1(H) be the largest integer d such that every (k−1)-element set of vertices of H belongs to at least d edges of H. Further, let t(k, n) be the smallest integer t such that every k-uniform hypergraph on n vertices and with δk−1(H) ≥ t contains a perfect matching. The parameter t(k, n) has been completely determined for all k and large n divisible by k by Rödl, Ruciński, and Szemerédi in [Perfect matchings in large uniform hypergraphs with large minimum collective degree, submitted]. The values of t(k, n) are very close to n/2−k. In fact, the function t(k, n) = n/2 − k + cn,k, where cn,k ∈ {3/2, 2, 5/2, 3} depends on the parity of k and n. The aim of this short note is to present a simple proof of an only slightly weaker bound: t(k, n) ≤ n/2+k/4. Our argument is based on an idea used in a recent paper of Aharoni, Georgakopoulos, and Sprüssel.