Cycles with almost linearly many chords

We prove that constant minimum degree already forces cycles with almost linearly many chords. Specifically, every graph $G$ with $\delta (G)\ge C$ contains a cycle of length $\mathrm{\ell }\ge 4$ with $\mathrm{\Omega }(\mathrm{\ell }\mathrm{/}{\log }^C\mathrm{\ell })$ chords for some absolute constant $C>0$. This is the first result showing that a constant-degree condition yields an unbounded – indeed nearly linear – number of chords, placing our bound within a polylogarithmic factor of the Chen–Erdős–Staton conjecture. It also gives a strong affirmative conclusion in the direction of a recent question of Dvořák, Martins, Thomassé, and Trotignon asking whether constant-degree graphs must contain cycles whose chord counts grow with their length.