Disjoint representability of sets and their complements

被引:10
|
作者
Balogh, J [1 ]
Keevash, P
Sudakov, B
机构
[1] Ohio State Univ, Dept Math Sci, Columbus, OH 43210 USA
[2] Princeton Univ, Dept Math, Princeton, NJ 08544 USA
[3] Inst Adv Study, Princeton, NJ 08540 USA
基金
美国国家科学基金会;
关键词
set systems; trace; extremal problems;
D O I
10.1016/j.jctb.2005.02.005
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
For a hypergraph H and a set S, the trace of W on S is the set of all intersections of edges of R with S. We will consider forbidden trace problems, in which we want to find the largest hypergraph W that does not contain some list of forbidden configurations as traces, possibly with some restriction on the number of vertices or the size of the edges in H. In this paper we will focus on combinations of three forbidden configurations: the k-singleton [k]((1)), the k-co-singleton [k]((k-1)) and the k-chain C-k = {0, {1}, [1, 2],..., [1, k-1]}, where we write [k]((l)) for the set of all l-subsets of [k]={1,..., k}. Our main topic is hypergraphs with no k-singleton or k-co-singleton trace. We obtain an exact result in the case k = 3, both for uniform and non-uniform hypergraphs, and classify the extremal examples. In the general case, we show that the number of edges in the largest r-uniform hypergraph with no k-singleton or k-co-singleton trace is of order r(k-2). By contrast, Frankl and Pach showed that the number of edges in the largest r-uniform hypergraph with no k-singleton trace is of order r(k-1). We also give a very short proof of the recent result of Balogh and Bollobas that there is a finite bound on the number of sets in any hypergraph without a k-singleton, k-co-singleton or k-chain trace, independently of the number of vertices or the size of the edges. (C) 2005 Elsevier Inc. All rights reserved.
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页码:12 / 28
页数:17
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