A note on non-jumping numbers

Let r ≥ 2 be an integer. A real number α ∈ [0,1) is a jump for r if there exists c > 0 such that for any ǫ > 0 and any integer m, m ≥ r, there exists an integer n0 such that any r-uniform graph with n > n0 vertices and density ≥ α + ǫ contains a subgraph with m vertices and density ≥ α+ c. It follows from a theorem of Erdýos and Stone that every α ∈ [0,1) is a jump for r = 2. Erdýos asked whether the same is true for r ≥ 3. In the paper ‘Hypergraphs do not jump’ (Combinatorica 4 � . � ,