Every q-edge-colouring of K_{2^q+1} contains a monochromatic odd cycle of length O(2^q/q^{1-o(1)}), the first bound of the form o(2^q).
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Monochromatic odd cycles in edge-coloured complete graphs
Every q-edge-colouring of K_{2^q+1} contains a monochromatic odd cycle of length O(2^q/q^{1-o(1)}), the first bound of the form o(2^q).