On \q\)-divisible Hypergraphs

Abstract

Let $H(V,E)$ be an $r$-uniform hypergraph. Let $A\subset V$ be a subset of vertices and define ${\mathrm{deg}}_H(A)=|\{e\in E:A\subset e\}|$.

We say that $H$ is $(k,m)$-divisible if for every $k$-subset $A$ of  $V(H)$, ${\mathrm{deg}}_H(A)\equiv 0(\mathrm{mod}m)$. (We assume that $1\le k<r$).

Given positive integers $r\ge 2$, $k\ge 1$ and $q$ a prime power, we prove that if $H$ is an $r$-uniform hypergraph and $|E|>(q-1)(\genfrac{}{}{0ex}{}{\mid V\mid }{k})$, then $H$ contains a nontrivial subhypergraph $F$ which is $(k,q)$-divisible.