A note on the jumping constant conjecture of Erdös

Let r>=2 be an integer. The real number @a@?[0,1] is a jump for r if there exists c>0 such that for every positive @e and every integer m>=r, every r-uniform graph with n>n"0(@e,m) vertices and at least (@a+@e)(nr) edges contains a subgraph with m vertices and at least (@a+c)(mr) edges. A result of Erdos, Stone and Simonovits implies that every @a@?[0,1) is a jump for r=2. For r>=3, Erdos asked whether the same is true and showed that every @a@?[0,r!r^r) is a jump. Frankl and Rodl gave a negative answer by showing that 1-1l^r^-^1 is not a jump for r if r>=3 and l>2r. Another well-known question of Erdos 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 52r!r^r is not a jump for r>=3. We also describe an infinite sequence of non-jumping numbers for r=3.