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- W2948629369 abstract "An {em ordered $r$-graph} is an $r$-uniform hypergraph whose vertex set is linearly ordered; it is $r$-{em interval-partite} if there are $r$ consecutive intervals such that each edge has one point in each interval. The basic observation of Erdős and Kleitman that every $r$-graph contains an $r$-partite subgraph with at least a constant (depending on $r$) proportion of edges cannot be extended to the setting of $r$-interval-partite subgraphs in ordered hypergraphs. However, we prove that one can obtain a relatively dense subgraph that has the following similar structure. An ordered $r$-graph is a {em split} hypergraph if there exist $r-1$ consecutive intervals in the ordering with one of them specified, such that every edge has two vertices in the specified interval and one vertex in every other interval. Our main result is a ``splitting theorem'' that for each $alpha > lfloor r/2 rfloor$, every $n$-vertex ordered $r$-graph with $d ,n^{alpha}$ edges has for some $mleq n$ an $m$-vertex split subgraph with $Omega(d, m^{alpha})$ edges. The restriction $alpha > lfloor r/2 rfloor$ is sharp. The splitting theorem has interesting applications to several extremal problems for ordered hypergraphs." @default.
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- W2948629369 date "2019-06-07" @default.
- W2948629369 modified "2023-09-27" @default.
- W2948629369 title "A splitting theorem for ordered hypergraphs" @default.
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