Erdős–Gould–Yuster–Oz conjecture on dense chordal subgraphs

A chordal subgraph of a graph is a subgraph with no induced cycle of length greater than three.

Erdős–Gould–Yuster–Oz conjecture. Any graph with $n$ vertices and more than $n^2/3$ edges contains a chordal subgraph with at least $8n/3-4$ edges. The complete tripartite graph with one additional edge shows that this would be sharp.

The conjecture seeks the sharp lower bound for a large chordal subgraph above the density threshold $n^2/3$. The source records only a weaker bound of $7n/3-6$ edges, so the conjecture is unresolved there.