arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1906.03342math.CO
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划分有序超图

Partitioning ordered hypergraphs

Zoltán F\" uredi, Tao Jiang, Alexandr Kostochka, Dhruv Mubayi, Jacques Verstraëte

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16 pages
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摘要：

{ \em序$r$ -图}是顶点集是线性序的$r$ -一致超图.给定$2\leq k\leq r$，一个有序$r$ -图$H$是{ \em interval} $k$ -{ \em partite}，如果在序中至少存在$k$不交区间，使得$H$的每条边与每个区间都有非空交集，并且包含在它们的并集中.我们的主要结果是：对每个$α> k - 1$和$d>0$，每个$n$ -顶点序的$r$ -图有$d \,n^α$-边，对于某些$m\leq n$，有一个$m$ -顶点区间的$k$ -部的子图有$Ω(d\, m^α)$-边.这是Erd \H os和Kleitman关于每个$r$ -图都包含一个边的比例为常数的$r$ -部子图的观察的有序$r$ -图的推广.限制$α> k-1$是尖锐的。我们还提出了应用程序的主要结果的几个极值问题的有序超图。

英文摘要：

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 $α> k - 1$ and $d>0$, every $n$-vertex ordered $r$-graph with $d \,n^α$ edges has for some $m\leq n$ an $m$-vertex interval $k$-partite subgraph with $Ω(d\, m^α)$ 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 $α> k-1$ is sharp. We also present applications of the main result to several extremal problems for ordered hypergraphs.

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