论文标题
存在彩虹匹配的合作条件
Cooperative conditions for the existence of rainbow matchings
论文作者
论文摘要
令$ k> 1 $，然后让$ \ Mathcal {f} $成为$ 2N+K-3 $非空的边缘的家族。如果$ \ Mathcal {f} $的每个$ k $成员的结合包含尺寸$ n $的匹配，则存在$ \ nathcal {f} $ - 大小$ n $的彩虹匹配。将$ 2N+K-3 $替换为$ 2N+K-2 $，结果也是$ k = 1 $的，并且可以通过拓扑和相对简单的组合参数证明（对于所有$ k $）。主要的努力是获得最后的$ 1 $，这使结果敏锐。
Let $k>1$, and let $\mathcal{F}$ be a family of $2n+k-3$ non-empty sets of edges in a bipartite graph. If the union of every $k$ members of $\mathcal{F}$ contains a matching of size $n$, then there exists an $\mathcal{F}$-rainbow matching of size $n$. Replacing $2n+k-3$ by $2n+k-2$, the result is true also for $k=1$, and it can be proved (for all $k$) both topologically and by a relatively simple combinatorial argument. The main effort is in gaining the last $1$, which makes the result sharp.
