On a conjecture of Erdős and Simonovits: Even cycles

Let F be a family of graphs. A graph is F-free if it contains no copy of a graph in F as a subgraph. A cornerstone of extremal graph theory is the study of the Turán number ex(n,F), the maximum number of edges in an F-free graph on n vertices. Define the Zarankiewicz number z(n,F) to be the maximum number of edges in an F-free bipartite graph on n vertices. Let Ck denote a cycle of length k, and let Ck denote the set of cycles C`, where 3 ≤ ` ≤ k and ` and k have the same parity. Erdős and Simonovits conjectured that for any family F consisting of bipartite graphs there exists an odd integer k such that ex(n,F ∪Ck) ∼ z(n,F). They proved this when F = {C4} by showing that ex(n, {C4, C5}) ∼ z(n,C4). In this paper, we extend this result by showing that if ` ∈ {2, 3, 5} and k > 2` is odd, then ex(n, C2`∪{Ck}) ∼ z(n, C2`). Furthermore, if k > 2`+ 2 is odd, then for infinitely many n we show that the extremal C2` ∪ {Ck}-free graphs are bipartite incidence graphs of generalized polygons. We observe that this exact result does not hold for any odd k < 2`, and furthermore the asymptotic result does not hold when (`, k) is (3, 3), (5, 3) or (5, 5). Our proofs make use of pseudorandomness properties of nearly extremal graphs that are of independent interest.