Hypergraphs do jump
Abstract
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 Erd os--Stone--Simonovits theorem implies that for r=2 every α∈ [0,1) is a jump. Erd os 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 R\"odl disproved this conjecture by giving a sequence of values of non-jumps for all r≥ 3. We use Razborov's flag algebra method 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 Tur\'an 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.