Unavoidable cycle lengths in graphs

An old conjecture of Erdos states that there exists an absolute constant c and a set S of density zero such that every graph of average degree at least c contains a cycle of length in S. In this paper, we prove this conjecture by showing that every graph of average degree at least ten contains a cycle of length in a prescribed set S satisfying | S ∧ {1, 2,..., n}| = O(n(0.99)). © 2005 Wiley Periodicals, Inc.