Short monochromatic odd cycles

TL;DR

This paper improves bounds on the length of the shortest monochromatic odd cycle in k-edge-coloured complete graphs, using algebraic combinatorics and approximation theory.

Contribution

It provides an exponential improvement over previous bounds on L(k), the minimal odd cycle length in k-edge-coloured complete graphs.

Findings

Established that L(k)=O(k^{3/2}2^{k/2})

Improved previous polynomial bounds exponentially

Applied algebraic combinatorics and approximation theory techniques

Abstract

It is easy to see that every $k$-edge-colouring of the complete graph on $2^k+1$ vertices contains a monochromatic odd cycle. In 1973, Erd\H{o}s and Graham asked to estimate the smallest $L(k)$ such that every $k$-edge-colouring of $K_{2^k+1}$ contains a monochromatic odd cycle of length at most $L(k)$. Recently, Gir\~ao and Hunter obtained the first nontrivial upper bound by showing that $L(k)=O(\frac{2^k}{k^{1-o(1)}})$, which improves the trivial bound by a polynomial factor. We obtain an exponential improvement by proving that $L(k)=O(k^{3/2}2^{k/2})$. Our proof combines tools from algebraic combinatorics and approximation theory.