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arXiv · 2412.07708

Monochromatic odd cycles in edge-coloured complete graphs

António Girão·Zach Hunter
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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 Erdős 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 $L(q)$.

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António Girão, Zach Hunter. 2024-12-10. Monochromatic odd cycles in edge-coloured complete graphs. https://arxiv.org/abs/2412.07708

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