AI 中文总结
研究埃尔德什 - 罗杰斯函数\(f_{s,s + 1}(n)\)，通过构造无\(K_{s + 1}\)图得出其上界，利用约雷特等人关于团色数的定理得出下界，最终证明\(f_{s,s + 1}(n) = \Theta(\sqrt{n \log n})\)。
AI 中文摘要
我们证明了对于每个\(s \geq 2\)，埃尔德什 - 罗杰斯函数\(f_{s,s + 1}(n)\)满足\(f_{s,s + 1}(n) = \Theta(\sqrt{n \log n})\)。具体而言，我们构造了一个\(n\)个顶点的无\(K_{s + 1}\)图，其中每至少\(C(s)\sqrt{n \log n}\)个顶点的集合包含一个\(K_s\)的副本，这给出了上界。匹配的下界由约雷特、米采克、里德和斯米德关于图的团色数的定理得出。
英文摘要
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.
Comments22 pages
