On the order of the classical Erdős–Rogers functions

IF 0.9 3区数学 Q2 MATHEMATICS Bulletin of the London Mathematical Society Pub Date : 2024-12-20 DOI:10.1112/blms.13214

Dhruv Mubayi, Jacques Verstraete

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Abstract

For an integer $n ⩾ 1$ , the Erdős–Rogers function $f_{s, s + 1} (n)$ is the maximum integer $m$ such that every $n$ -vertex $K_{s + 1}$ -free graph has a $K_{s}$ -free induced subgraph with $m$ vertices. It is known that for all $s ⩾ 3$ , $f_{s, s + 1} (n) = Ω (\sqrt{n \log n} / \sqrt{\log \log n})$ as $n \to \infty$ . In this paper, we show that for all $s ⩾ 3$ , there exists a constant $c_{s} > 0$ such that

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按照经典Erdős-Rogers函数的顺序

对于整数n小于1 $n \geqslant 1$， Erdős-Rogers函数f s，S + 1 (n) $f_{s,s+1}(n)$是最大整数m $m$使得每n个$n$ -顶点K s + 1 $K_{s+1}$自由图有一个K s $K_s$自由诱导子图，有m个$m$个顶点。已知对于所有s小于3 $s \geqslant 3$, f s，S + 1 (n) = Ω (n log n /Log (Log n) $f_{s,s+1}(n) = \Omega (\sqrt {n\log n}/\sqrt {\log \log n})$ = n→∞$n \rightarrow \infty$。在本文中，我们表明，对于所有s小于3 $s \geqslant 3$，存在一个常数c s ＆gt；0 $c_s > 0$这样

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