Rainbow matchings for 3-unifrom hypergraphs

K\"{u}hn, Osthus and Treglown and, independently, Khan proved that if $H$ is a $3$-uniform hypergraph with $n$ vertices such that $n\in 3\mathbb Z$ and large, and the minimum vertex degree of $H$ is greater than ${n-1\choose 2}-{2n/3\choose 2}$, then $H$ contains a perfect matching. Huang, Loh, and Sudakov showed that if, for $1\le i\le t$, where $t\binom{n}{k}-\binom{n-t+1}{k}$, then $\{F_1,\dots,F_t\}$ admits a rainbow matching. We show that for $n\in 3\Z$ sufficiently large, if, for $i\in \{1, \ldots, n/3\}$, ${F}_i\subseteq {[n]\choose 3}$ and $\delta_1({F}_i)>{n-1\choose 2}-{2n/3\choose 2}$, then $\{F_1,\dots, F_{n/3}\}$ admits a rainbow matching. This is joint work with Hongliang Lu and Xingxing Yu.