Partitioning Edge-Coloured Complete Symmetric Digraphs into Monochromatic Complete Subgraphs

TL;DR

This paper constructs a specific 2-colouring of an infinite complete symmetric digraph where monochromatic paths have zero density, and establishes covering and stability results for colourings with restricted path lengths.

Contribution

It provides the first example of a 2-colouring with zero density monochromatic paths and proves covering and stability theorems for multi-coloured infinite digraphs with path length restrictions.

Findings

Constructed a 2-colouring with zero density monochromatic paths.

Proved that graphs with no long monochromatic paths can be covered by a bounded number of monochromatic subgraphs.

Established a stability result showing the colouring is nearly unique under density constraints.

Abstract

Let $K_{\mathbb{N}}$ 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 $K_{\mathbb{N}}$ in which every monochromatic path has density~$0$. However, if we restrict the length of monochromatic paths in one colour, then no example as above can exist: We show that every $(r+1)$-edge-coloured complete symmetric digraph (of arbitrary infinite cardinality) containing no directed paths of edge-length $\ell_i$ for any colour $i\leq r$ can be covered by $\prod_{i\leq r} \ell_i$ pairwise disjoint monochromatic complete symmetric digraphs in colour $r+1$. Furthermore, we present a stability version for the countable case of the latter result: We prove that the edge-colouring is uniquely determined on a large subgraph, as soon as the upper density of monochromatic paths in colour $r+1$ is bounded by $\prod_{i\in [r]}\frac{1}{\ell_i}$.