Abstract The t -color Ramsey problem for hypergraph matchings was settled by the well-known result of Alon, Frankl and Lovasz (answering a conjecture of Erdős). This result was the last step in a chain of special cases most notably Lovasz’s solution to Kneser’s problem. We proposed an extension of the Erdős problem: for given 1 ≤ s ≤ t , what is the maximum number of vertices that can be covered by a matching having at most s colors in every t -coloring of the edges of the complete graph K n (or hypergraph K n r ). We revisit the first unknown case, r = 2 , s = 2 , t = 4 , where we conjectured that in every 4-coloring of K n there is a bicolored matching covering at least ⌊ 3 n ∕ 4 ⌋ vertices. We prove that this is true asymptotically by applying a recent twist of a standard application of the Regularity method: instead of lifting a (bicolored) matching of the reduced graph to regular cluster pairs, we lift a (bicolored) basic 2-matching, a subgraph whose connected components are edges and odd cycles. To find the bicolored basic 2-matching with at least ⌊ 3 n ∕ 4 ⌋ vertices in every 4-coloring of K n we use Tutte’s minimax formula.

中文翻译：

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彩色完全图和超图中颜色很少的匹配

摘要 超图匹配的 t-color Ramsey 问题由 Alon、Frankl 和 Lovasz 的著名结果（回答 Erdős 的猜想）解决。这个结果是一系列特殊情况的最后一步，最著名的是 Lovasz 对 Kneser 问题的解决方案。我们提出了 Erdős 问题的扩展：对于给定的 1 ≤ s ≤ t ，在完整图 K n 的边的每个 t 着色中，最多具有 s 种颜色的匹配可以覆盖的最大顶点数是多少？ （或超图 K nr ）。我们重新审视第一个未知情况 r = 2 , s = 2 , t = 4 ，我们推测在 K n 的每个 4-着色中都有一个双色匹配覆盖至少 ⌊ 3 n ∕ 4 ⌋ 个顶点。我们通过应用正则方法的标准应用的最新变化来证明这是渐近的：我们不是将简化图的（双色）匹配提升到常规集群对，而是提升（双色）基本 2-匹配，即连接组件为边和奇数圈的子图。为了在 K n 的每个 4-着色中找到至少具有 ⌊ 3 n ∕ 4 ⌋ 个顶点的双色基本 2-匹配，我们使用 Tutte 的极小极大公式。