 Bipartite stability from few odd cycles

Fix an odd cycle length. If the graph has edge density at least $1/2-ϵ$ and sufficiently few copies of that cycle, odd-cycle copies from positive triangle density forces few triangles in a graph. The triangle removal lemma deletes at most $ϵ\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ edges, after which the clique-free edit bound applies with two parts. Absorbing its linear rounding error into the density margin gives the displayed bound for large order.