  We say that $\alpha\in [0,1)$ is a jump for an integer $r\geq 2$ if there
exists $c(\alpha)>0$ such that for all $\epsilon >0 $ and all $t\geq 1$ any
$r$-graph with $n\geq n_0(\alpha,\epsilon,t)$ vertices and density at least
$\alpha+\epsilon$ contains a subgraph on $t$ vertices of density at least
$\alpha+c$. The Erd\H os--Stone--Simonovits theorem implies that for $r=2$
every $\alpha\in [0,1)$ is a jump. Erd\H os showed that for all $r\geq 3$,
every $\alpha\in [0,r!/r^r)$ is a jump. Moreover he made his famous "jumping
constant conjecture" that for all $r\geq 3$, every $\alpha \in [0,1)$ is a
jump. Frankl and R\"odl 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 $\alpha \in [0.2299,0.2316)$ is a jump. We also
give an improved upper bound for the Tur\'an density of
$K_4^-=\{123,124,134\}$: $\pi(K_4^-)\leq 0.2871$. This in turn implies that for
$r=3$ every $\alpha \in [0.2871,8/27)$ is a jump.
