Hypergraphs do jump

We say that $α\in [0,1)$ is a jump for an integer $r\geq 2$ if there exists $c(α)>0$ such that for all $ε>0$ and all $t\geq 1$ any $r$-graph with $n\geq n_0(α,ε,t)$ vertices and density at least $α+ε$ contains a subgraph on $t$ vertices of density at least $α+c$. The Erd\H os--Stone--Simonovits theorem implies that for $r=2$ every $α\in [0,1)$ is a jump. Erd\H os showed that for all $r\geq 3$, every $α\in [0,r!/r^r)$ is a jump. Moreover he made his famous "jumping constant conjecture" that for all $r\geq 3$, every $α\in [0,1)$ is a jump. Frankl and Rödl disproved this conjecture by giving a sequence of values of non-jumps for all $r\geq 3$. 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 $r\geq 3$ in the interval $[r!/r^r,1)$. To be precise we show that for $r=3$ every $α\in [0.2299,0.2316)$ is a jump. We also give an improved upper bound for the Turán density of $K_4^-=\{123,124,134\}$: $π(K_4^-)\leq 0.2871$. This in turn implies that for $r=3$ every $α\in [0.2871,8/27)$ is a jump.

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