  A real number $\alpha\in [0, 1)$ is a jump for an integer $r\ge 2$ if there
exists $c>0$ such that no number in $(\alpha , \alpha + c)$ can be the Tur\'an
density of a family of $r$-uniform graphs. A classical result of Erd\H os and
Stone \cite{ES} implies that that every number in $[0, 1)$ is a jump for $r=2$.
Erd\H os \cite{E64} also showed that every number in $[0, r!/r^r)$ is a jump
for $r\ge 3$ and asked whether every number in $[0, 1)$ is a jump for $r\ge 3$.
Frankl and R\"odl \cite{FR84} gave a negative answer by showing a sequence of
non-jumps for every $r\ge 3$. After this, Erd\H os modified the question to be
whether $\frac{r!}{r^r}$ is a jump for $r\ge 3$? What's the smallest non-jump?
Frankl, Peng, R\"odl and Talbot \cite{FPRT} showed that ${5r!\over 2r^r}$ is a
non-jump for $r\ge 3$. Baber and Talbot \cite{BT0} showed that every
$\alpha\in[0.2299, 0.2316)\cup [0.2871, \frac{8}{27})$ is a jump for $r=3$.
Pikhurko \cite{Pikhurko2} showed that the set of all possible Tur\'an densities
of $r$-uniform graphs has cardinality of the continuum for $r\ge 3$. However,
whether $\frac{r!}{r^r}$ is a jump for $r\ge 3$ remains open, and
$\frac{5r!}{2r^r}$ has remained the known smallest non-jump for $r\ge 3$. In
this paper, we give a smaller non-jump by showing that ${54r!\over 25r^r}$ is a
non-jump for $r\ge 3$. Furthermore, we give infinitely many irrational
non-jumps for every $r\ge 3$.
