Rainbow cycles in properly edge-colored graphs

We prove that every properly edge-colored $n$-vertex graph with average degree at least $100(\log n{)}^2$ contains a rainbow cycle, improving upon $(\log n{)}^{2+o(1)}$ bound due to Tomon. We also prove that every properly colored $n$-vertex graph with at least ${10}^5{k}^2{n}^{1+1\mathrm{/}k}$ edges contains a rainbow $2k$-cycle, which improves the previous bound $2^{c{k}^2}{n}^{1+1\mathrm{/}k}$ 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.