Year: 2026
Type: preprint
Abstract: We show that the Erdős-Rogers function $f_{s,s+1}(n)$ satisfies $$f_{s,s+1}(n) = Θ( \sqrt{n \log n} )$$ for every $s \ge 2$. More precisely, we construct a $K_{s+1}$-free graph on $n$ vertices in which every set of at least $C(s)\sqrt{n \log n}$ vertices contains a copy of $K_s$ for some constant $C(s)$, which implies the upper bound. The matching lower bound follows from a theorem of Joret, Micek, Reed and Smid on the clique chromatic number of a graph.
Source: arXiv (Cornell University)
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7169757151
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