Density of monochromatic infinite subgraphs II

Corsten, Jan, DeBiasio, Louis and McKenney, Paul (2025) Density of monochromatic infinite subgraphs II. Forum of Mathematics Sigma, 13.

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Abstract

In 1967, Gerencsér and Gyárfás [16] proved a result which is considered the starting point of graph-Ramsey theory: In every 2-coloring of Kn, there is a monochromatic path on ⌈(2n+1)/3⌉ vertices, and this is best possible. There have since been hundreds of papers on graph-Ramsey theory with some of the most important results being motivated by a series of conjectures of Burr and Erdős [2, 3] regarding the Ramsey numbers of trees (settled in [31]), graphs with bounded maximum degree (settled in [5]), and graphs with bounded degeneracy (settled in [23]). In 1993, Erdős and Galvin [13] began the investigation of a countably infinite analogue of the Gerencsér and Gyárfás result: What is the largest d such that in every 2-coloring of KN there is a monochromatic infinite path with upper density at least d? Erdős and Galvin showed that 2/3≤d≤8/9, and after a series of recent improvements, this problem was finally solved in [7] where it was shown that d=12+√8)/17. This paper begins a systematic study of quantitative countably infinite graph-Ramsey theory, focusing on infinite analogues of the Burr-Erdős conjectures. We obtain some results which are analogous to what is known in finite case, and other (unexpected) results which have no analogue in the finite case.

Item Type:	Article
Additional Information:	© 2025 The Author(s)
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	27 May 2025 11:24
Last Modified:	27 May 2025 11:24
URI:	http://eprints.lse.ac.uk/id/eprint/128179

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