A note on improved bounds for hypergraph rainbow matching problems

Abstract: A natural question, inspired by the famous Ryser-Brualdi-Stein Conjecture, is to determine the largest positive integer $g(r,n)$ such that every collection of $n$ matchings, each of size $n$, in an $r$-partite $r$-uniform hypergraph contains a rainbow matching of size $g(r,n)$. The parameter $g&#39;(r,n)$ is defined identically with the exception that the host hypergraph is not required to be $r$-partite. In this note, we improve the best known lower bounds on $g&#39;(r,n)$ for all $r \geq 4$ and the upper bounds on $g(r,n)$ for all $r \geq 3$, provided $n$ is sufficiently large. More precisely, we show that if $r\ge3$ then $$\frac{2n}{r+1}-\Theta_r(1)\le g&#39;(r,n)\le g(r,n)\le n-\Theta_r(n<sup>{1-\frac{1}{r}}).$$ Interestingly, while it has been conjectured that $g(2,n)=g&#39;(2,n)=n-1$, our results show that if $r\ge3$ then $g(r,n)$ and $g&#39;(r,n)$ are bounded away from $n$ by a function which grows in $n$. We also prove analogous bounds for the related problem where we are interested in the smallest size $s$ for which any collection of $n$ matchings of size $s$ in an ($r$-partite) $r$-uniform hypergraph contains a rainbow matching of size $n$.