TY - JOUR

T1 - Ramsey upper density of infinite graph factors

AU - Balogh, József

AU - Lamaison, Ander

N1 - Funding Information:
The research presented here was conducted during a visit by the second author to the University of Illinois Urbana-Champaign. We would like to thank the Berlin Mathematical School for funding this trip. We thank the referee for carefully reading the manuscript.
Funding Information:
Acknowledgments. The first author’s research is partially supported by NSF Grant DMS-1764123, Arnold O. Beckman Research Award (UIUC Campus Research Board RB 18132), the Langan Scholar Fund (UIUC), and the Simons Fellowship.
Funding Information:
The second author’s research is currently supported by the MUNI Award in Science and Humanities of the Grant Agency of Masaryk University. Previous research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATHC (EXC-2046/1, project ID: 390685689).
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 -