A note on the Turán function of even cycles

Abstract

The Tur´an function ex(n, F) is the maximum number of edges in an F-free graph on n vertices. The question of estimating this function for F = C2k, the cycle of length 2k, is one of the central open questions in this area that goes back to the 1930s. We prove that ex(n,C2k) ≤ (k − 1) n1+1/k + 16(k − 1)n, improving the previously best known general upper bound of Verstra¨ete [Combin. Probab. Computing 9 (2000), 369–373] by a factor 8 + o(1) when n � k.