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 $\epsilon >0$ and any integer $m,m\ge r$, any $r$-uniform graph with $n>n_0(\epsilon ,m)$ vertices and at least α+ \varepsilon)\binom{n}{r}$edgescontainsasubgraphwith$m$verticesandatleast$(α+c)\binom{m}{r}$edges,where$c=c(α)$doesnotdependon$\varepsilon$and$m$.ItfollowsfromatheoremofErd\stackrel{\backslash H}{o}s,StoneandSimonovitsthatevery$α\in [0,1)$isajumpfor$r=2$.Erd\stackrel{\backslash H}{o}saskedwhetherthesameistruefor$r \geq 3$.However,FranklandR\ddot{o}dlgaveanegativeanswerbyshowingthat$1-\frac{1}{\ell^{r-1}}$isnotajumpfor$r$if$r \geq 3$and$\ell >2r$.Penggavemoresequencesofnon-jumpingnumbersfor$r=4$and$r\geq 3$.However,therearealsoalotofunknownsondeterminingwhetheranumberisajumpfor$r \geq 3$.FollowingasimilarapproachasthatofFranklandR\ddot{o}dl,wegiveseveralsequencesofnon-jumpingnumbersfor$r=5$,andextendoneoftheresultstoevery$r \geq 5$, which generalize the above results.

29 pages