Partitioning ordered hypergraphs

An {\em ordered $r$-graph} is an $r$-uniform hypergraph whose vertex set is linearly ordered. Given $2\leq k\leq r$, an ordered $r$-graph $H$ is {\em interval} $k$-{\em partite} if there exist at least $k$ disjoint intervals in the ordering such that every edge of $H$ has nonempty intersection with each of the intervals and is contained in their union. Our main result implies that for each $\alpha > k - 1$ and $d>0$, every $n$-vertex ordered $r$-graph with $d \,n^{\alpha}$ edges has for some $m\leq n$ an $m$-vertex interval $k$-partite subgraph with $\Omega(d\, m^{\alpha})$ edges. This is an extension to ordered $r$-graphs of the observation by Erd\H os and Kleitman that every $r$-graph contains an $r$-partite subgraph with a constant proportion of the edges. The restriction $\alpha > k-1$ is sharp. We also present applications of the main result to several extremal problems for ordered hypergraphs.