Upper bounds found for perfect matchings in bipartite hypergraphs

What the study found: The authors prove an upper bound on the number of A-perfect matchings in uniform bipartite hypergraphs with small maximum codegree. They also derive bounds for the number of transversals in certain order-n Latin squares and for proper q-edge-colorings of certain regular hypergraphs.
Why the authors say this matters: The study suggests these counting bounds apply to combinatorial structures such as Latin squares and edge-colorings. The findings indicate a connection between matchings in hypergraphs and limitations on how many such configurations can exist.
What the researchers tested: The paper studies bipartite hypergraphs with bipartition (A, B), where every edge has exactly one vertex in A, and A-perfect matchings, which saturate every vertex in A. The authors prove an upper bound for uniform hypergraphs with small maximum codegree and use it to obtain further counting results.
What worked and what didn't: The upper bound was established for A-perfect matchings under the stated small-codegree condition. Using it, the authors show that there exist order-n Latin squares with at most (n/e^{2.117})^n transversals when n is odd and n ≡ 0 mod 3, and that k-uniform D-regular hypergraphs on n vertices have at most ((1+o(1))q/e^k)^{Dn/k} proper q-edge-colorings when q = (1+o(1))D and the maximum codegree is o(q).
What to keep in mind: The abstract gives only the conditions under which the bounds hold, including small maximum codegree and the stated assumptions on n, q, D, and k. It does not describe limitations beyond those scope conditions.

Key points

• The paper proves an upper bound on the number of A-perfect matchings in uniform bipartite hypergraphs.
• The bound applies when the hypergraph has small maximum codegree.
• The authors use the result to bound transversals in some order-n Latin squares.
• They also bound the number of proper q-edge-colorings in certain k-uniform D-regular hypergraphs.
• For odd n with n ≡ 0 mod 3, the abstract states a bound of at most (n/e^{2.117})^n transversals.