Non-jumping Numbers for 5-Uniform Hypergraphs

Let $\ell$ and $r$ be integers. A real number $\alpha \in [0,1)$ is a jump for $r$ if for any $\varepsilon > 0$ and any integer $m,\ m \geq r$, any $r$-uniform graph with $n > n_0(\varepsilon,m)$ vertices and at least \alpha+ \varepsilon)\binom{n}{r}$edges contains a subgraph with$m$vertices and at least$(\alpha +c)\binom{m}{r}$edges, where$c=c(\alpha)$does not depend on$\varepsilon$and$m$. It follows from a theorem of Erd\H{o}s, Stone and Simonovits that every$\alpha \in [0,1)$is a jump for$r=2$. Erd\H{o}s asked whether the same is true for$r \geq 3$. However, Frankl and R\"{o}dl gave a negative answer by showing that$1-\frac{1}{\ell^{r-1}}$is not a jump for$r$if$r \geq 3$and$\ell >2r$. Peng gave more sequences of non-jumping numbers for$r=4$and$r\geq 3$. However, there are also a lot of unknowns on determining whether a number is a jump for$r \geq 3$. Following a similar approach as that of Frankl and R\"{o}dl, we give several sequences of non-jumping numbers for$r=5$, and extend one of the results to every$r \geq 5$, which generalize the above results.