Monochromatic odd cycles in edge-coloured complete graphs

TL;DR

This paper establishes a new upper bound on the smallest monochromatic odd cycle length in any q-edge-colouring of a complete graph with 2^q+1 vertices, advancing understanding of edge-colouring Ramsey theory.

Contribution

It provides the first non-trivial upper bound on L(q), the minimal odd cycle length guaranteed in q-edge-colourings of K_{2^q+1}.

Findings

L(q)=O(2^q / q^{1-o(1)})

First non-trivial upper bound on L(q)

Advances understanding of monochromatic odd cycles in edge-coloured complete graphs

Abstract

It is easy to see that every $q$-edge-colouring of the complete graph on $2^q+1$ vertices must contain a monochromatic odd cycle. A natural question raised by Erd\H{o}s and Graham in $1973$ asks for the smallest $L(q)$ such that every $q$-edge-colouring of $K_{2^q+1}$ must contain a monochromatic odd cycle of length at most $L(q)$. In here, we show that $L(q)=O\left(\frac{2^q}{q^{1-o(1)}}\right)$ giving the first non-trivial upper bound on $L(q)$.