Ramsey numbers of trails and circuits

TL;DR

This paper establishes a nearly optimal lower bound on the length of monochromatic trails or circuits guaranteed in any two-colouring of a complete graph, advancing understanding of Ramsey numbers for such structures.

Contribution

It provides the first asymptotically tight bound for the Ramsey numbers of trails and circuits in two-coloured complete graphs.

Findings

Monochromatic trails or circuits of length at least 2n^2/9 + o(n^2) exist in any two-colouring.

The bound is asymptotically best possible.

Advances the understanding of Ramsey numbers for paths and cycles.

Abstract

We show that every two-colouring of the edges of the complete graph $K_n$ contains a monochromatic trail or circuit of length at least $2n^2/9 +o(n^2)$, which is asymptotically best possible.