Let r 2 be an integer. A real number ∈ [0, 1) is a jump for r if there is a constant c > 0 such that for any > 0 and any integer m where m r , there exists an integer n0 such that any r-uniform graph with n > n0 vertices and density + contains a subgraph with m vertices and density + c. It follows from a fundamental theorem of Erdős and Stone that every ∈ [0, 1) is a jump for r = 2. Erdős asked whether the same is true for r 3. Frankl and Rödl gave a negative answer by showing some non-jumping… CONTINUE READING