  How many edges in an $n$-vertex graph will force the existence of a cycle
with as many chords as it has vertices? Almost 30 years ago, Chen, Erd\H{o}s
and Staton considered this question and showed that any $n$-vertex graph with
$2n^{3/2}$ edges contains such a cycle. We significantly improve this old bound
by showing that $\Omega(n\log^8n)$ edges are enough to guarantee the existence
of such a cycle.
  Our proof exploits a delicate interplay between certain properties of random
walks in almost regular expanders. We argue that while the probability that a
random walk of certain length in an almost regular expander is self-avoiding is
very small, one can still guarantee that it spans many edges (and that it can
be closed into a cycle) with large enough probability to ensure that these two
events happen simultaneously.
