Ramsey upper density of infinite graph factors

József Balogh, Ander Lamaison

Research output: Contribution to journalArticlepeer-review


The study of upper density problems on Ramsey theory was initiated by Erdos and Galvin in 1993 in the particular case of the infinite path, and by DeBiasio and McKenney in general. In this paper, we are concerned with the following problem: Given a fixed finite graph F, what is the largest value of λ such that every 2-edge-coloring of the complete graph on N contains a monochromatic infinite F -factor whose vertex set has upper density at least λ? Here we prove a new lower bound for this problem. For some choices of F, including cliques and odd cycles, this new bound is sharp because it matches an older upper bound. For the particular case where F is a triangle, we also give an explicit lower bound of 1 - 1/√7 = 0.62203..., improving the previous best bound of 3/5.

Original languageEnglish (US)
Pages (from-to)171-184
Number of pages14
JournalIllinois Journal of Mathematics
Issue number1
StatePublished - Apr 2023

ASJC Scopus subject areas

  • General Mathematics


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