On counting triangulations in d dimensions
Computational Geometry: Theory and Applications
On the number of simplicial complexes in Rd
Computational Geometry: Theory and Applications
Proof of a conjecture of Mader, Erdös and Hajnal on topological complete subgraphs
European Journal of Combinatorics
The extremal function for complete minors
Journal of Combinatorial Theory Series B
Graph Minors. XX. Wagner's conjecture
Journal of Combinatorial Theory Series B - Special issue dedicated to professor W. T. Tutte
Homological Connectivity Of Random 2-Complexes
Combinatorica
Homological connectivity of random k-dimensional complexes
Random Structures & Algorithms
Minors of simplicial complexes
Discrete Applied Mathematics
On multiplicative λ-approximations and some geometric applications
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
On laplacians of random complexes
Proceedings of the twenty-eighth annual symposium on Computational geometry
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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.