Entropy Bounds for Perfect Matchings in Bipartite Hypergraphs

A hypergraph is $\mathit{\text{bipartite}}\mathit{\text{with}}\mathit{\text{bipartition}}(A,B)$ if every edge has exactly one vertex in $A$, and a matching in such a hypergraph is $A\mathit{\text{-perfect}}$ if it saturates every vertex in $A$. We prove an upper bound on the number of $A$-perfect matchings in uniform hypergraphs with small maximum codegree. Using this result, we prove that there exist order-$n$ Latin squares with at most $(n\mathrm{/}{e}^{2.117}{)}^n$ transversals when $n$ is odd and $n\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}3)$. We also show that $k$-uniform $D$-regular hypergraphs on $n$ vertices have at most $((1+o(1))q\mathrm{/}{e}^k{)}^{Dn\mathrm{/}k}$ proper $q$-edge-colorings when $q=(1+o(1))D$ and the maximum codegree is $o(q)$.