Improved Bound on Vertex Degree Version of Erd\H{o}s Matching Conjecture

Abstract

For a $k$-uniform hypergraph $H$, let $\delta_1(H)$ denote the minimum vertex degree of $H$, and $\nu(H)$ denote the size of the largest matching in $H$. In this paper, we show that for any $k\geq 3$ and $\beta>0$, there exists an integer $n_0(\beta,k)$ such that for positive integers $n\geq n_0$ and $m\leq (\frac{k}{2(k-1)}-\beta)\frac{n}{k}$, if $H$ is an $n$-vertex $k$-graph with $\delta_1(H)>{{n-1}\choose {k-1}}-{{n-m}\choose {k-1}},$ then $\nu(H)\geq m$. This improves upon earlier results of Bollob\'{a}s, Daykin and Erd\H{o}s (1976) for the range $n> 2k^3(m+1)$ and Huang and Zhao (2017) for the range $n\geq 3k^2 m$.