An infinite graph is highly connected if the complement of any subgraph of smaller size is connected. We consider weaker versions of Ramsey’s Theorem asserting that in any coloring of the edges of a complete graph there exist large highly connected subgraphs all of whose edges are colored by the same color.
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The first author would like to thank Chris Lambie-Hanson for valuable discussion of the consistency strengths of failures of \(\lambda\)-stationary \(\square(\mu)\) sequences to exist. The authors would like to thank the referee for a prompt, alert, and intelligent reading. This paper is number 1157 in the third authors publication list.
The research of the second author was supported by a PAPIIT grant No. IN100317 and CONACyT grant No. A1-S-16164.
The research of the third author was partially supported by NSF grant 136974 and by ERC grant 338821.
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Bergfalk, J., Hrušák, M. & Shelah, S. Ramsey theory for highly connected monochromatic subgraphs. Acta Math. Hungar. 163, 309–322 (2021). https://doi.org/10.1007/s10474-020-01058-x
Key words and phrases
- Ramsey theory
- k-connected graph
- highly connected graph
- Mahlo cardinal
- weakly compact cardinal
Mathematics Subject Classification