Chen, Erdos, and Staton asked in 1996 how many edges are required in an n-vertex graph to guarantee a cycle with as many chords as vertices. The best current bound, due to Draganic, Methuku, Munha Correia, and Sudakov shows that average degree (log n)^8 already suffices.

In this talk, I will show that constant average degree is enough to guarantee a cycle on \ell vertices with at least Ω{ \ell \over {log^{C }(\ell)} } chords answering a question of Dvorak, Martins, Thomasse, and Trotignon asking whether constant-degree graphs must contain cycles whose chord counts grow with their length.

This is joint work with Nemanja Draganic,