Generating non-jumping numbers recursively

Let r⩾2r⩾2 be an integer. A real number α∈[0,1)α∈[0,1) is a jump for r   if there is a constant c>0c>0 such that for any ε>0ε>0 and any integer m   where m⩾rm⩾r, there exists an integer n0n0 such that any r  -uniform graph with n>n0n>n0 vertices and density ⩾α+ε⩾α+ε contains a subgraph with m   vertices and density ⩾α+c⩾α+c. It follows from a fundamental theorem of Erdős and Stone that every α∈[0,1)α∈[0,1) is a jump for r=2r=2. Erdős asked whether the same is true for r⩾3r⩾3. Frankl and Rödl gave a negative answer by showing some non-jumping numbers for every r⩾3r⩾3. In this paper, we provide a recursive formula to generate more non-jumping numbers for every r⩾3r⩾3 based on the current known non-jumping numbers.