We say that α ∈ [0, 1) is a jump for an integer r ≥ 2 if there exists c(α) > 0 such that for all ϵ > 0 and all t ≥ 1, any r-graph with n ≥ n0(α, ϵ, t) vertices and density at least α + ϵ contains a subgraph on t vertices of density at least α + c.
The Erdős–Stone–Simonovits theorem [4, 5] implies that for r = 2, every α ∈ [0, 1) is a jump. Erdős [3] showed that for all r ≥ 3, every α ∈ [0, r!/rr) is a jump. Moreover he made his famous ‘jumping constant conjecture’, that for all r ≥ 3, every α ∈ [0, 1) is a jump. Frankl and Rödl [7] disproved this conjecture by giving a sequence of values of non-jumps for all r ≥ 3.
We use Razborov's flag algebra method [9] to show that jumps exist for r = 3 in the interval [2/9, 1). These are the first examples of jumps for any r ≥ 3 in the interval [r!/rr, 1). To be precise, we show that for r = 3 every α ∈ [0.2299, 0.2316) is a jump.
We also give an improved upper bound for the Turán density of K4− = {123, 124, 134}: π(K4−) ≤ 0.2871. This in turn implies that for r = 3 every α ∈ [0.2871, 8/27) is a jump.