TY - JOUR

T1 - Ramsey upper density of infinite graph factors

AU - Balogh, József

AU - Lamaison, Ander

N1 - Publisher Copyright:
© 2023 by the University of Illinois Urbana-Champaign.

PY - 2023/4

Y1 - 2023/4

N2 - 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.

AB - 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.

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U2 - 10.1215/00192082-10450499

DO - 10.1215/00192082-10450499

M3 - Article

AN - SCOPUS:85159664123

SN - 0019-2082

VL - 67

SP - 171

EP - 184

JO - Illinois Journal of Mathematics

JF - Illinois Journal of Mathematics

IS - 1

ER -