Matchings in n-partite n-graphs (Q1820172)

Let \(H=(V,E)\) be an n-partite n-graph such that V is partitioned into set \(V_ 1,v_ 2,...,v_ n\). For any subset A of \(V_ 1\) let \(K(A)=\{(v_ 2,...,v_ n):\) \((a,v_ 2,...,v_ n)\in E\) for some \(a\in A\}\) and let \(\nu\) (H) denote the size of the largest matching in H, where H is any hypergraph. Furthermore, let \(H_ A\) be the (n-1)-graph \((V_ 2\cup...\cup V_ n\), K(A)). A fractional matching is a real valued function f on E satisfying \(\sum_{v\in E}f(e)\leq 1\) for any \(v\in V\). Also, \(\nu^*(H)\) denotes max\(\sum_{e}f(e)\) over all fractional matchings f in H. The author proves that if \(\nu (H_ A)\geq (n-1)(| A| -1)+1\) for every subset A of \(V_ 1\), then \(\nu^*(H)=| V_ 1|\). The author conjectures that if \(\nu (H_ A)\geq (n-1)(| A| -1)+1\) for every subset A of \(V_ 1\), then \(V_ 1\) is matchable.