A note on the jumping constant conjecture of Erdös

Let r 2 be an integer. The real number α ∈ [0,1] is a jump for r if there exists c > 0 such that for every positive and every integer m r , every r-uniform graph with n > n0( ,m) vertices and at least (α + )(nr) edges contains a subgraph with m vertices and at least (α + c)(mr ) edges. A result of Erdős, Stone and Simonovits implies that every α ∈ [0,1) is a jump for r = 2. For r 3, Erdős asked whether the same is true and showed that every α ∈ [0, r! rr ) is a jump. Frankl and Rödl gave a negative answer by showing that 1 − 1 lr−1 is not a jump for r if r 3 and l > 2r . Another well-known question of Erdős is whether r! rr is a jump for r 3 and what is the smallest non-jumping number. In this paper we prove that 5 2 r! rr is not a jump for r 3. We also describe an infinite sequence of non-jumping numbers for r = 3. © 2006 Elsevier Inc. All rights reserved.