Submission date: 25 March 2021

Abstract:

We prove a version of the strong Taylor's conjecture for stable graphs: if G is a stable graph whose chromatic number is strictly greater than ℶ_2(ℵ_0) then G contains all finite subgraphs of Sh_n(ω) and thus has elementary extensions of unbounded chromatic number. This completes the picture from our previous work. The main new model theoretic ingredient is a generalization of the classical construction of Ehrenfeucht-Mostowski models to an infinitary setting, giving a new characterization of stability.

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Full text arXiv 2103.13931: pdf, ps.