A Stability Theorem for Maximal $C_{2k+1}$-free Graphs

For any positive integer $k$, we show that every maximal $C_{2k+1}$-free graph with at least $n^2\mathrm{/}4-o(n^{3\mathrm{/}2})$ edges contains an induced complete bipartite subgraph on $(1-o(1))n$ vertices. We also show that this is best possible.