INFINITE STABLE GRAPHS WITH LARGE CHROMATIC NUMBER

Yatir Halevi; Itay Kaplan; Saharon Shelah

doi:10.1090/tran/8570

INFINITE STABLE GRAPHS WITH LARGE CHROMATIC NUMBER

Yatir Halevi, Itay Kaplan, Saharon Shelah

Ben-Gurion University of the Negev, The Hebrew University of Jerusalem

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that if G = (V, E) is an ω-stable (respectively, superstable) graph with χ(G) > ℵ₀ (respectively, 2^ℵ⁰ ) then G contains all the finite subgraphs of the shift graph Sh_n(ω) for some n. We prove a variant of this theorem for graphs interpretable in stationary stable theories. Furthermore, if G is ω-stable with U(G) ≤ 2 we prove that n ≤ 2 suffices.

Original language	American English
Pages (from-to)	1767-1799
Number of pages	33
Journal	Transactions of the American Mathematical Society
Volume	375
Issue number	3
DOIs	https://doi.org/10.1090/tran/8570
State	Published - 1 Jan 2022

Keywords

Chromatic number
Stable graphs
Taylor’s conjecture

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

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10.1090/tran/8570

Cite this

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abstract = "We prove that if G = (V, E) is an ω-stable (respectively, superstable) graph with χ(G) > ℵ0 (respectively, 2ℵ0 ) then G contains all the finite subgraphs of the shift graph Shn(ω) for some n. We prove a variant of this theorem for graphs interpretable in stationary stable theories. Furthermore, if G is ω-stable with U(G) ≤ 2 we prove that n ≤ 2 suffices.",

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N2 - We prove that if G = (V, E) is an ω-stable (respectively, superstable) graph with χ(G) > ℵ0 (respectively, 2ℵ0 ) then G contains all the finite subgraphs of the shift graph Shn(ω) for some n. We prove a variant of this theorem for graphs interpretable in stationary stable theories. Furthermore, if G is ω-stable with U(G) ≤ 2 we prove that n ≤ 2 suffices.

AB - We prove that if G = (V, E) is an ω-stable (respectively, superstable) graph with χ(G) > ℵ0 (respectively, 2ℵ0 ) then G contains all the finite subgraphs of the shift graph Shn(ω) for some n. We prove a variant of this theorem for graphs interpretable in stationary stable theories. Furthermore, if G is ω-stable with U(G) ≤ 2 we prove that n ≤ 2 suffices.

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KW - Taylor’s conjecture

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INFINITE STABLE GRAPHS WITH LARGE CHROMATIC NUMBER

Abstract

Keywords

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