  Aharoni and Howard, and, independently, Huang, Loh, and Sudakov proposed the
following rainbow version of Erd\H{o}s matching conjecture: For positive
integers $n,k,m$ with $n\ge km$, if each of the families $F_1,\ldots,
F_m\subseteq {[n]\choose k}$ has size more than $\max\{\binom{n}{k} -
\binom{n-m+1}{k}, \binom{km-1}{k}\}$, then there exist pairwise disjoint
subsets $e_1,\dots, e_m$ such that $e_i\in F_i$ for all $i\in [m]$. We prove
that there exists an absolute constant $n_0$ such that this rainbow version
holds for $k=3$ and $n\geq n_0$. We convert this rainbow matching problem to a
matching problem on a special hypergraph $H$. We then combine several existing
techniques on matchings in uniform hypergraphs: find an absorbing matching $M$
in $H$; use a randomization process of Alon et al. to find an almost regular
subgraph of $H-V(M)$; and find an almost perfect matching in $H-V(M)$. To
complete the process, we also need to prove a new result on matchings in
3-uniform hypergraphs, which can be viewed as a stability version of a result
of {\L}uczak and Mieczkowska and might be of independent interest.
