A note on the jumping constant conjecture of Erdős

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 ϵ and every integer m ⩾ r , every r -uniform graph with n > n 0 ( ϵ , m ) vertices and at least ( α + ϵ ) ( n r ) edges contains a subgraph with m vertices and at least ( α + c ) ( m r ) 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 ! r r ) is a jump. Frankl and Rödl gave a negative answer by showing that 1 − 1 l r − 1 is not a jump for r if r ⩾ 3 and l > 2 r . Another well-known question of Erdős is whether r ! r r is a jump for r ⩾ 3 and what is the smallest non-jumping number. In this paper we prove that 5 2 r ! r r is not a jump for r ⩾ 3 . We also describe an infinite sequence of non-jumping numbers for r = 3 .