On the order of the classical Erdős–Rogers functions

IF 0.8 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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引用次数: 0

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