For any subset $A \subseteq \mathbb{N}$, we define its upper density to be $\limsup_{ n \rightarrow \infty } |A \cap \{ 1, \dotsc, n \}| / n$. We prove that every $2$-edge-colouring of the complete graph on $\mathbb{N}$ contains a monochromatic infinite path, whose vertex set has upper density at least $(9 + \sqrt{17})/16 \approx 0.82019$. This improves on results of Erdős and Galvin, and of DeBiasio and McKenney.
@article{10_37236_7758,
author = {Allan Lo and Nicol\'as Sanhueza-Matamala and Guanghui Wang},
title = {Density of monochromatic infinite paths},
journal = {The electronic journal of combinatorics},
year = {2018},
volume = {25},
number = {4},
doi = {10.37236/7758},
zbl = {1402.05074},
url = {http://geodesic.mathdoc.fr/articles/10.37236/7758/}
}
TY - JOUR
AU - Allan Lo
AU - Nicolás Sanhueza-Matamala
AU - Guanghui Wang
TI - Density of monochromatic infinite paths
JO - The electronic journal of combinatorics
PY - 2018
VL - 25
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/7758/
DO - 10.37236/7758
ID - 10_37236_7758
ER -
%0 Journal Article
%A Allan Lo
%A Nicolás Sanhueza-Matamala
%A Guanghui Wang
%T Density of monochromatic infinite paths
%J The electronic journal of combinatorics
%D 2018
%V 25
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/7758/
%R 10.37236/7758
%F 10_37236_7758
Allan Lo; Nicolás Sanhueza-Matamala; Guanghui Wang. Density of monochromatic infinite paths. The electronic journal of combinatorics, Tome 25 (2018) no. 4. doi: 10.37236/7758
