Matching of given sizes in hypergraphs

Abstract

For all integers $k,d$ such that $k\ge 3$ and $k/2\le d\le k-1$, let $n$ be a sufficiently large integer {\rm(}which may not be divisible by $k${\rm)} and let $s\le \lfloor n/k\rfloor -1$. We show that if $H$ is a $k$-uniform hypergraph on $n$ vertices with ${\delta }_d(H)>\left(\genfrac{}{}{0ex}{}{n-d}{k-d}\right)-\left(\genfrac{}{}{0ex}{}{n-d-s+1}{k-d}\right)$, then $H$ contains a matching of size $s$. This improves a recent result of Lu, Yu, and Yuan and also answers a question of Kühn, Osthus, and Townsend. In many cases, our result can be strengthened to $s\le \lfloor n/k\rfloor$, which then covers the entire possible range of $s$. On the other hand, there are examples showing that the result does not hold for certain $n,k,d$ and $s=\lfloor n/k\rfloor$.

arXiv admin note: text overlap with arXiv:1507.02362

References in corpus (2)

• Fractional and integer matchings in uniform hypergraphs
• Near Perfect Matchings in $k$-uniform Hypergraphs