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Minors in random and expanding hypergraphs
 in 27th SoCG’, ACM
, 2011
"... We introduce a new notion of minors for simplicial complexes (hypergraphs), socalled 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, ..."
Abstract

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We introduce a new notion of minors for simplicial complexes (hypergraphs), socalled 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 LinialMeshulam model X k (n, p). By definition, such a complex has n vertices, a complete (k − 1)dimensional skeleton, and every possible kdimensional 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 X k (n, p) asymptotically almost surely contains K k t (the complete kdimensional complex on t vertices) as a homological minor. As corollary, the threshold for (topological) embeddability of X k (n, p) into R 2k 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 Tverbergtype problems, where instead of continuous maps without doubly covered image points (embeddings), we consider maps without qfold covered image points.
Bounding Helly Numbers via Betti Numbers *
"... Abstract We show that very weak topological assumptions are enough to ensure the existence of a Hellytype theorem. More precisely, we show that for any nonnegative integers b and d there exists an integer h (b, d) such that the following holds. If F is a finite family of subsets of R d such thatβ ..."
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Abstract We show that very weak topological assumptions are enough to ensure the existence of a Hellytype theorem. More precisely, we show that for any nonnegative integers b and d there exists an integer h (b, d) such that the following holds. If F is a finite family of subsets of R d such thatβ i ( G) ≤ b for any G F and every 0 ≤ i ≤ d/2 − 1 then F has Helly number at most h (b, d). Hereβ i denotes the reduced Z 2 Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these d/2 first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological nonembeddability results with a Ramseybased approach to build, given an arbitrary simplicial complex K, some wellbehaved chain map C * (K) → C * (R d ). Both techniques are of independent interest.