Abstract:We show that there is an absolute constant $c>0$ such that the following holds. For every $n > 1$, there is a 5-uniform hypergraph on at least $2^{2^{cn^{1/4}}}$ vertices with independence number at most $n$, where every set of 6 vertices induces at most 3 edges. The double exponential growth rate for the number of vertices is sharp. By applying a stepping-up lemma established by the first two authors, analogous sharp results are proved for $k$-uniform hypergraphs. This answers the penultimate open case of a conjecture in Ramsey theory posed by Erdős and Hajnal in 1972.

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中文翻译：

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关于 Erdős-Hajnal 超图 Ramsey 问题的注释

摘要：我们证明存在一个绝对常数$c>0$，使得以下成立。对于每$n > 1$，在至少$2^{2^{cn^{1/4}}}$个顶点上存在一个5-uniform超图，其独立数最多为$n$，其中每组6个顶点最多诱导 3 条边。顶点数量的双指数增长率是急剧的。通过应用前两位作者建立的递推引理，对于$k$-均匀超图也证明了类似的尖锐结果。这回答了 Erdős 和 Hajnal 在 1972 年提出的 Ramsey 理论猜想的倒数第二个公开案例。

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