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Hypergraphs do not jump. (English) Zbl 0663.05047
Let $$G=G(V,E)$$ be a graph on $$n$$ vertices with $$V=$$ vertex set and $$E=$$ edge set $$\subset V\times V)$$. The ratio of the number edges in the graph to the total number possible is called the density of $$G$$, i.e. $$d(G)=| E| /\binom{n}{2}$$. There is an unexpected “jump” in the density of a subgraph versus the graph itself. It follows by a theorem of Erdős, Stone and Simonovitis that for any positive integer $$m\geq 2$$, real $$0\leq \alpha \leq 1$$, and $$n$$ sufficiently large, any graph on $$n$$ vertices having density greater than $$\alpha$$ contains a subgraph on m vertices having density greater than $$\alpha +c$$, where $$c$$ is some fixed, positive constant not depending on $$m$$ or $$n$$. For example, in the class of complete $$\ell$$-partite graphs whose partition classes are of size $$k$$ there exist subgraphs which are complete $$m$$-points with densities $$=1$$ that exceed $$d(G)$$ by more than $$c=1/(\ell +1)$$ for arbitrary $$k>\ell$$ and $$2\leq m\leq \ell$$ (since in this case $$d(G)=(k\ell +k)/(k\ell -1)).$$
In this interesting paper, the authors extend the problem to include $$r$$-uniform hypergraphs – graphs whose “edges” are $$r$$-element subsets of $$V$$ (in this more general setting, $$d(G)=| E| /\binom{n}{r}$$. The precise definition for jump used here is: a real number $$0\leq \alpha \leq 1$$ is a jump for $$r$$ provided that for any positive $$c$$ and any integer $$m\geq r$$, an $$r$$-uniform hypergraph with $$n>n_ 0(\varepsilon,m)$$ vertices and density at least $$\alpha +\varepsilon$$ contains a subgraph on $$m$$ vertices with density at least $$\alpha +c$$, where $$c=c(\alpha)$$ does not depend on $$\varepsilon$$ and $$m$$. By use of the Lagrange function on graphs and an analysis of complete $$\ell$$-partite $$r$$-uniform graphs, the authors prove that the numbers $$1-1/\ell^{r-1}$$ for $$\ell =2r+1,2r+2,\dots$$ are not jumps if $$r\geq 3$$. This settles a question of Erdős who has offered a \$ 1,000 prize for the answer.
Reviewer: D. Kay

##### MSC:
 05C65 Hypergraphs
##### Keywords:
density; hypergraphs; jump
Full Text:
##### References:
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