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.

我们证明每个平均度数至少为\(32(\log 5n)^2\)的正确边缘着色的n顶点图都包含彩虹循环，改进了\((\log n)^{2+o( 1)}\)由于托蒙而受到约束。我们还证明，每个具有至少\(10^5 k^3 n^{1+1/k}\)边的正确边缘着色的n顶点图都包含彩虹 2 k循环，这改进了先前的界限\ (2^{ck^2}n^{1+1/k}\)由 Janzer 获得。我们使用同态不等式和不平衡正则化引理的方法还提供了一种简单的方法来证明偶循环的 Erdős-Simonovits 过饱和定理，这可能具有独立的意义。