Perfect Matchings in Hypergraphs and the Erd\H{o}s matching conjecture

Abstract

We prove a new upper bound for the minimum $d$-degree threshold for perfect matchings in $k$-uniform hypergraphs when $d<k/2$. As a consequence, this determines exact values of the threshold when $0.42k \le d < k/2$ or when $(k,d)=(12,5)$ or $(17,7)$. Our approach is to give an upper bound on the Erd\H{o}s Matching Conjecture and convert the result to the minimum $d$-degree setting by an approach of K\"uhn, Osthus and Townsend. To obtain exact thresholds, we also apply a result of Treglown and Zhao.