Non-jumping Numbers for 5-Uniform Hypergraphs

Abstract: Let $\ell$ and $r$ be integers. A real number $\alpha \in [0,1)$ is a jump for $r$ if for any $\varepsilon &gt; 0$ and any integer $m,\ m \geq r$, any $r$-uniform graph with $n &gt; n_0(\varepsilon,m)$ 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$$. However, Frankl and R\&quot;{o}dl gave a negative answer by showing that$$1-\frac{1}{\ell^{r-1}}</sup>isnotajumpfor</sup> is not a jump for rif if r \geq 3and and \ell >2r.Penggavemoresequencesofnon−jumpingnumbersfor. Peng gave more sequences of non-jumping numbers for r=4and and r\geq 3.However,therearealsoalotofunknownsondeterminingwhetheranumberisajumpfor. However, there are also a lot of unknowns on determining whether a number is a jump for r \geq 3.FollowingasimilarapproachasthatofFranklandR&quot;odl,wegiveseveralsequencesofnon−jumpingnumbersfor. Following a similar approach as that of Frankl and R\&quot;{o}dl, we give several sequences of non-jumping numbers for r=5,andextendoneoftheresultstoevery, and extend one of the results to every r \geq 5$, which generalize the above results.