Abstract
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.
Highlights
A 2-edge-colouring of a graph G is an assignment of 2 colours, red and blue, to each edge of G
Given an arbitrary 2-edge-colouring of Kn, what is the size of the largest monochromatic path contained as a subgraph? This was answered by Gerencser and Gyarfas [7], who proved that every 2-edge-coloured Kn contains a monochromatic path of length at least 2n/3
We focus on the case of monochromatic paths in 2-edge-coloured complete graphs
Summary
A 2-edge-colouring of a graph G is an assignment of 2 colours, red and blue, to each edge of G. This was answered by Gerencser and Gyarfas [7], who proved that every 2-edge-coloured Kn contains a monochromatic path of length at least 2n/3 By a classical result of Ramsey Theory, any 2-edge-colouring of KN contains a monochromatic infinite complete graph, and a monochromatic infinite path P. Rado [8] showed that in every r-edge-coloured KN there are r monochromatic paths, of distinct colours, which partition the vertex set. This immediately implies that every 2-edge-coloured KN contains an infinite monochromatic path P with d(P ) 1/2.
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