Rainbow Cycles in Properly Edge-Colored Graphs

We prove that every properly edge-colored n-vertex graph with average degree at least \(32(\log 5n)^2\) contains a rainbow cycle, improving upon the \((\log n)^{2+o(1)}\) bound due to Tomon. We also prove that every properly edge-colored n-vertex graph with at least \(10^5 k^3 n^{1+1/k}\) edges contains a rainbow 2k-cycle, which improves the previous bound \(2^{ck^2}n^{1+1/k}\) obtained by Janzer. Our method using homomorphism inequalities and a lopsided regularization lemma also provides a simple way to prove the Erdős–Simonovits supersaturation theorem for even cycles, which may be of independent interest.