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.
| Original language | English |
|---|---|
| Pages (from-to) | 1767-1799 |
| Number of pages | 33 |
| Journal | Transactions of the American Mathematical Society |
| Volume | 375 |
| Issue number | 3 |
| DOIs | |
| State | Published - 2022 |
Bibliographical note
Funding Information:Received by the editors August 4, 2020, and, in revised form, September 13, 2020, and July 21, 2021. 2020 Mathematics Subject Classification. Primary 03C45; Secondary 05C15. Key words and phrases. Chromatic number, stable graphs, Taylor’s conjecture. For the first author, this research was supported by the Israel Science Foundation (grant No. 181/16) and the Kreitman foundation fellowship. For the second author, this research (grants no. 1533/14 and 1254/18) was supported by the Israel Science Foundation. The third author was supported by the Israel Science Foundation grant no: 1838/19 and the European Research Council grant 338821. Paper no. 1196 in the third author’s publication list.
Publisher Copyright:
© 2021 American Mathematical Society.
Keywords
- Chromatic number
- Stable graphs
- Taylor’s conjecture