Turán numbers of bipartite graphs plus an odd cycle

For an odd integer k  , let Ck={C3,C5,…,Ck}Ck={C3,C5,…,Ck} denote the family of all odd cycles of length at most k   and let CC denote the family of all odd cycles. Erdős and Simonovits [10] conjectured that for every family FF of bipartite graphs, there exists k   such that ex(n,F∪Ck)∼ex(n,F∪C)ex(n,F∪Ck)∼ex(n,F∪C) as n→∞n→∞. This conjecture was proved by Erdős and Simonovits when F={C4}F={C4}, and for certain families of even cycles in [14]. In this paper, we give a general approach to the conjecture using Scott's sparse regularity lemma. Our approach proves the conjecture for complete bipartite graphs K2,tK2,t and K3,3K3,3: we obtain more strongly that for any odd k⩾5k⩾5,ex(n,F∪{Ck})∼ex(n,F∪C) and we show further that the extremal graphs can be made bipartite by deleting very few edges. In contrast, this formula does not extend to triangles – the case k=3k=3 – and we give an algebraic construction for odd t⩾3t⩾3 of K2,tK2,t-free C3C3-free graphs with substantially more edges than an extremal K2,tK2,t-free bipartite graph on n vertices. Our general approach to the Erdős–Simonovits conjecture is effective based on some reasonable assumptions on the maximum number of edges in an m by n   bipartite FF-free graph.