37 research outputs found

    General Position Subsets and Independent Hyperplanes in d-Space

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    Erd\H{o}s asked what is the maximum number α(n)\alpha(n) such that every set of nn points in the plane with no four on a line contains α(n)\alpha(n) points in general position. We consider variants of this question for dd-dimensional point sets and generalize previously known bounds. In particular, we prove the following two results for fixed dd: - Every set HH of nn hyperplanes in Rd\mathbb{R}^d contains a subset SHS\subseteq H of size at least c(nlogn)1/dc \left(n \log n\right)^{1/d}, for some constant c=c(d)>0c=c(d)>0, such that no cell of the arrangement of HH is bounded by hyperplanes of SS only. - Every set of cqdlogqcq^d\log q points in Rd\mathbb{R}^d, for some constant c=c(d)>0c=c(d)>0, contains a subset of qq cohyperplanar points or qq points in general position. Two-dimensional versions of the above results were respectively proved by Ackerman et al. [Electronic J. Combinatorics, 2014] and by Payne and Wood [SIAM J. Discrete Math., 2013].Comment: 8 page

    Positive independence densities of finite rank countable hypergraphs are achieved by finite hypergraphs

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    The independence density of a finite hypergraph is the probability that a subset of vertices, chosen uniformly at random contains no hyperedges. Independence densities can be generalized to countable hypergraphs using limits. We show that, in fact, every positive independence density of a countably infinite hypergraph with hyperedges of bounded size is equal to the independence density of some finite hypergraph whose hyperedges are no larger than those in the infinite hypergraph. This answers a question of Bonato, Brown, Kemkes, and Pra{\l}at about independence densities of graphs. Furthermore, we show that for any kk, the set of independence densities of hypergraphs with hyperedges of size at most kk is closed and contains no infinite increasing sequences.Comment: To appear in the European Journal of Combinatorics, 12 page
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