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Minors in random and expanding hypergraphs

Published: 13 June 2011 Publication History

Abstract

We introduce a new notion of minors for simplicial complexes (hypergraphs), so-called homological minors. Our motivation is to propose a general approach to attack certain extremal problems for sparse simplicial complexes and the corresponding threshold problems for random complexes. In this paper, we focus on threshold problems. The basic model for random complexes is the Linial-Meshulam model Xk(n,p). By definition, such a complex has n vertices, a complete (k-1)-dimensional skeleton, and every possible k-dimensional simplex is chosen independently with probability p. We show that for every k,t ≥ 1, there is a constant C=C(k,t) such that for p ≥ C/n, the random complex Xk(n,p) asymptotically almost surely contains Kkt (the complete k-dimensional complex on t vertices) as a homological minor. As corollary, the threshold for (topological) embeddability of Xk(n,p) into R2k is at p=Θ(1/n).
The method can be extended to other models of random complexes (for which the lower skeleta are not necessarily complete) and also to more general Tverberg-type problems, where instead of continuous maps without doubly covered image points (embeddings), we consider maps without q-fold covered image points.

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    cover image ACM Conferences
    SoCG '11: Proceedings of the twenty-seventh annual symposium on Computational geometry
    June 2011
    532 pages
    ISBN:9781450306829
    DOI:10.1145/1998196
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    Published: 13 June 2011

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    Author Tags

    1. embeddings
    2. hypergraphs
    3. minors
    4. random complexes
    5. simplicial complexes
    6. thresholds

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    SoCG '11
    SoCG '11: Symposium on Computational Geometry
    June 13 - 15, 2011
    Paris, France

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