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

TL;DR

This paper establishes new upper bounds for perfect matchings in hypergraphs with specific degree conditions, solving parts of the Erdős matching conjecture and determining exact thresholds for certain parameters.

Contribution

It provides a novel upper bound for the minimum degree threshold for perfect matchings in hypergraphs when d<k/2, advancing understanding of the Erdős matching conjecture.

Findings

New upper bounds for minimum d-degree thresholds

Exact threshold values for specific (k,d) pairs

Application of existing results to determine thresholds

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.