A NOTE ON THE TURAN FUNCTION OF EVEN CYCLES

The Turan 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 = C-2k, 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, C-2k) <= (k - 1)n(1+1/k) + 16(k - 1)n, improving the previously best known general upper bound of Verstraete [Combin. Probab. Computing 9 (2000), 369-373] by a factor 8 + o(1) when n >> k.