For every k, the density of odd alternating paths P^A_{2k+1} in any edge-coloured graph is at most k^k(k+1)^{k+1}(2k+1)^{-2k-1}.
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Maximizing Alternating Paths via Entropy
For every k, the density of odd alternating paths P^A_{2k+1} in any edge-coloured graph is at most k^k(k+1)^{k+1}(2k+1)^{-2k-1}.