Monochromatic Paths in the Complete Symmetric Infinite Digraph

TL;DR

This paper constructs a 2-coloring of the complete symmetric infinite digraph where all monochromatic paths have zero density, and establishes density bounds for paths avoiding certain lengths in one color.

Contribution

It answers a question by constructing a specific coloring and proves a density bound for paths in the second color under constraints.

Findings

Existence of a 2-coloring with all monochromatic paths of density 0.

Density bounds for paths avoiding certain lengths in one color.

Construction method for such colorings.

Abstract

Let $\vec{K}_{\mathbb{N}}$ be the complete symmetric digraph on the positive integers. Answering a question of DeBiasio and McKenney, we construct a 2-colouring of the edges of $\vec{K}_{\mathbb{N}}$ in which every monochromatic path has density 0. On the other hand, we show that, in every colouring that does not have a directed path with $r$ edges in the first colour, there is directed path in the second colour with density at least $\frac1r$.