AI 中文总结
研究广义Erdős–Rogers问题，证明对于r≥3且满足条件的超图F和G，存在对数上界，改进了He和Nie的猜想，并恢复了Ramsey下界。
AI 中文摘要
设\(F\)和\(G\)为\(r\)-一致超图，并令\(f_{F,G}(n)\)为最大整数\(m\)，使得每个\(n\)顶点\(G\)-自由\(r\)-图包含一个\(m\)顶点的导出\(F\)-自由子图。我们证明，如果\(r\ge3\)，\(F\)非空，\(G\)是\(2\)-紧连通的，且不存在从\(G\)到\(F\)的同态，则\[ f_{F,G}(n)\le C(\log n)^{\beta_F}, \qquad \beta_F= \max_{\substack{\emptyset\ne P\subseteq\partial_2F}} \frac{e(P)}{v(P)-1}. \]对于\(r=3\)，这证实了He和Nie关于紧连通\(3\)-图的一个猜想，将其之前的界中的指数\( \max_{\substack{\emptyset\ne P\subseteq\partial_2F}} \frac{e(P)+1}{v(P)-1} \)改进为\(\beta_F\)。当\(F=K_r^r\)时，我们的结果恢复了Ramsey下界\(r(G,K_n^r)\ge 2^{\Omega(n^{2/r})}\)，只要\(G\)是\(2\)-紧连通且非\(r\)-部图。
英文摘要
Let \(F\) and \(G\) be \(r\)-uniform hypergraphs, and let \(f_{F,G}(n)\) be the largest integer \(m\) such that every \(n\)-vertex \(G\)-free \(r\)-graph contains an induced \(F\)-free subgraph on \(m\) vertices. We prove that, for \(r\ge3\) and \(2\le k\le r-1\), if \(F\) is nonempty, \(G\) is \(k\)-tightly connected, and there is no homomorphism from \(G\) to \(F\) (that is, \(G\not\to F\)), then \[ f_{F,G}(n)\le C(\log n)^{β_F^{(k)}}, \qquad β_F^{(k)}= \max_{\emptyset\ne P\subseteq\partial_kF} \frac{e(P)}{v(P)-1}. \] The case \(r=3\) of our result resolves a conjecture of He and Nie. As a consequence, we obtain the Ramsey lower bound \(r(G,K_n^r)\ge2^{Ω\bigl(n^{(r-1)/\binom rk}\bigr)}\) for every \(k\)-tightly connected non-\(r\)-partite \(r\)-graph \(G\). This extends a result of Conlon, Fox, Gunby, He, Mubayi, Suk, Verstraëte and Yu from the \(3\)-uniform setting.
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