Infinite Cliques in Simple and Stable Graphs

Yatir Halevi; Itay Kaplan; Saharon Shelah

doi:10.2140/mt.2025.4.231

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arXiv · 2408.05605

Infinite Cliques in Simple and Stable Graphs

Yatir Halevi·Itay Kaplan·Saharon Shelah

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Abstract

Suppose that $G$ is a graph of cardinality $\mu^+$ with chromatic number $\chi(G)\geq \mu^+$. One possible reason that this could happen is if $G$ contains a clique of size $\mu^+$. We prove that this is indeed the case when the edge relation is stable. When $G$ is a random graph (which is simple but not stable), this is not true. But still if in general the complete theory of $G$ is simple, $G$ must contain finite cliques of unbounded sizes.

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BibTeX RIS

Yatir Halevi, Itay Kaplan, Saharon Shelah. 2024-08-10. Infinite Cliques in Simple and Stable Graphs. https://doi.org/10.2140/mt.2025.4.231

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