Partitioning ordered hypergraphs

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