AI 中文总结
该研究通过超图容器和概率构造，证明了超图埃尔洛斯-罗杰斯函数的上界，并改进了相关问题的现有结果。
AI 中文摘要
对于整数\(k\leq s<t\)，令\(f^{(k)}_{s,t}(n)\)表示最大整数\(m\)，使得每个\(n\)顶点的不含\(K_t^{(k)}\)的\(k\) - 图包含一组\(m\)个顶点，且不包含\(K_s^{(k)}\)的副本。我们通过证明对于每个固定的\(s\geq4\)，\(f^{(4)}_{s,s + 1}(n)=(\log n)^{o(1)}\)，对康伦、福克斯和苏达科夫的一个问题给出了肯定答案。关键输入是一个新的\(3\) - 均匀估计：对于每个固定的\(s\geq3\)，\(f^{(3)}_{s,s + 1}(n)=O(\frac{\log n}{\log\log n})\)。证明结合了超图容器和概率构造。此外，对于每个固定的\(k\geq5\)，存在常数\(C_k>0\)，使得\(f^{(k)}_{k + 1,k + 2}(n)\leq\exp\left(C_k\frac{\log_{(k - 2)} n}{\log_{(k - 1)} n}\right)\)。
英文摘要
For integers \(k\le s<t\), the hypergraph Erdős--Rogers function \(f^{(k)}_{s,t}(n)\) is the largest integer \(m\) such that every \(n\)-vertex \(K_t^{(k)}\)-free \(k\)-graph contains a set of \(m\) vertices spanning no copy of \(K_s^{(k)}\). We prove that, for every fixed \(s\ge4\), \[ f^{(4)}_{s,s+1}(n)=(\log n)^{o(1)}, \] thereby resolving a problem posed by Conlon, Fox and Sudakov. The key input is a new \(3\)-uniform estimate: for every fixed \(s\ge3\), \(f^{(3)}_{s,s+1}(n)=O(\frac{\log n}{\log\log n})\), which improves the logarithmic upper bound of Dudek and Mubayi. The proof develops a probabilistic pair-coloring construction based on a robust auxiliary palette and hypergraph containers. As a further consequence, we obtain \(f^{(k)}_{k+1,k+2}(n)=(\log_{(k-3)} n)^{o(1)}\) for every fixed \(k\ge5\), making substantial progress towards a conjecture of Mubayi and Suk.
CommentsThis version incorporates several detailed refinements. 20 pages
