发表机构
Institute of Mathematics, EPFL（洛桑联邦理工学院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了Erdős-Rogers函数$f_{\ell,s}(n)$的渐近上界$n^{\frac{\ell}{2s-\ell}+o(1)}$，填补了Sudakov下界与Bradač上界之间的差距，并推广了Bradač关于非对角Ramsey数的构造。
AI 中文摘要
Erdős-Rogers函数$f_{\ell,s}(n)$是在每个$n$个顶点的$K_s$-free图中必然存在的$K_\ell$-free诱导子图的最大阶数。虽然当$s=\ell+1$时该函数已被充分理解，但$s$远大于$\ell$的情况仍然悬而未决。Sudakov的一个长期下界指出$f_{\ell,s}(n)\geq n^{\frac{\ell}{2s}+O_\ell(s^{-2})}$，而Bradač最近的结果表明$f_{\ell,s}(n)\leq n^{\frac{\ell-1}{s-1}+o(1)}$。在本文中，我们通过证明$f_{\ell,s}(n)= n^{\frac{\ell}{2s}+O_\ell(s^{-2})}$来渐近地填补这一差距。更精确地，我们证明对于所有$2\leq \ell<s$，有$f_{\ell,s}(n)\leq n^{\frac{\ell}{2s-\ell}+o(1)}$。我们的证明基于Bradač最近关于非对角Ramsey数的紧构造，该构造可视为我们结果中$\ell=2$的情形。
英文摘要
The Erdős-Rogers function $f_{\ell,s}(n)$ is the largest order of a $K_\ell$-free induced subgraph guaranteed to exist in every $K_s$-free graph on $n$ vertices. While this function is well understood for $s=\ell+1$, the case where $s$ is much larger than $\ell$ has remained wide open. A long-standing lower bound of Sudakov states that $f_{\ell,s}(n)\geq n^{\frac{\ell}{2s}+O_\ell(s^{-2})}$, while a recent result of Bradač shows that $f_{\ell,s}(n)\leq n^{\frac{\ell-1}{s-1}+o(1)}$. In this paper, we close this gap asymptotically by proving that $f_{\ell,s}(n)= n^{\frac{\ell}{2s}+O_\ell(s^{-2})}$. More precisely, we prove that for all $2\leq \ell<s$, we have $f_{\ell,s}(n)\leq n^{\frac{\ell}{2s-\ell}+o(1)}$. Our proof builds on Bradač's recent tight construction for off-diagonal Ramsey numbers, which can be viewed as the $\ell=2$ case of our result.
Comments28 pages
