Infinite Stable Graphs With Large Chromatic Number

Yatir Halevi; Itay Kaplan; Saharon Shelah

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arXiv · 2007.12139

Infinite Stable Graphs With Large Chromatic Number

Yatir Halevi·Itay Kaplan·Saharon Shelah

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Abstract

We prove that if $G=(V,E)$ is an $\omega$-stable (respectively, superstable) graph with $\chi(G)>\aleph_0$ (respectively, $2^{\aleph_0}$) then $G$ contains all the finite subgraphs of the shift graph $\text{Sh}_n(\omega)$ for some $n$. We prove a variant of this theorem for graphs interpretable in stationary stable theories. Furthermore, if $G$ is $\omega$-stable with $\mathrm{U}(G)\leq 2$ we prove that $n\leq 2$ suffices.

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BibTeX RIS

Yatir Halevi, Itay Kaplan, Saharon Shelah. 2020-07-23. Infinite Stable Graphs With Large Chromatic Number. https://arxiv.org/abs/2007.12139

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