This birthday note gives a “non-asymptotic” version of our earlier result with G. N. Sárközy and Szemerédi [3], in which Endre had the lion’s share. A hypergraph H with vertex set V defines the shadow graph G(H) whose vertex set is V and whose edge set is the set of pairs of V that are covered by some hyperedge of H . An edge coloring C of H defines a multicoloring, the shadow coloring $$C^{\prime}$$ C ′ on G(H) , by assigning all colors of C to an edge xy of G(H) that appear on some edge of H containing $$\{x, y\}$$ { x , y } . A matching in a graph is a set of pairwise disjoint edges. A matching in a graph is perfect if it covers all vertices of the graph. I show that in every ( r −1)-coloring C of the complete r -uniform hypergraph $$K^{r}_{n}$$ K n r there is a monochromatic perfect matching in the shadow coloring $$C^{\prime}$$ C ′ (assuming n ≥ r ≥ 2 and n is even).

中文翻译：

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阴影颜色的完美搭配

这个生日笔记给出了我们早期与 GN Sárközy 和 Szemerédi [3] 的结果的“非渐近”版本，其中 Endre 占了最大份额。顶点集为 V 的超图 H 定义了阴影图 G(H)，其顶点集为 V，边集是由 H 的某个超边覆盖的 V 对的集合。H 的边缘着色 C 定义了多色，即 G(H) 上的阴影着色 $$C^{\prime}$$ C ′ ，通过将 C 的所有颜色分配给 G(H) 的边 xy，这些颜色出现在某些H 的边包含 $$\{x, y\}$$ { x , y } 。图中的匹配是一组成对不相交的边。如果图中的匹配覆盖了图的所有顶点，则匹配是完美的。