The extremal number of cycles with all diagonals

In 1975, Erdős asked the following natural question: What is the maximum number of edges that an $n$-vertex graph can have without containing a cycle with all diagonals? Erdős observed that the upper bound $O(n^{5\mathrm{/}3})$ holds since the complete bipartite graph $K_{3,3}$ can be viewed as a cycle of length six with all diagonals. In this paper, we resolve this old problem. We prove that there exists a constant $C$ such that every $n$-vertex with $C{n}^{3\mathrm{/}2}$ edges contains a cycle with all diagonals. Since any cycle with all diagonals contains cycles of length four, this bound is best possible using well-known constructions of graphs without a four-cycle based on finite geometry. Among other ideas, our proof involves a novel lemma about finding an `almost-spanning’ robust expander which might be of independent interest.