6 research outputs found

    On Topological Minors in Random Simplicial Complexes

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    For random graphs, the containment problem considers the probability that a binomial random graph G(n,p)G(n,p) contains a given graph as a substructure. When asking for the graph as a topological minor, i.e., for a copy of a subdivision of the given graph, it is well-known that the (sharp) threshold is at p=1/np=1/n. We consider a natural analogue of this question for higher-dimensional random complexes Xk(n,p)X^k(n,p), first studied by Cohen, Costa, Farber and Kappeler for k=2k=2. Improving previous results, we show that p=Θ(1/n)p=\Theta(1/\sqrt{n}) is the (coarse) threshold for containing a subdivision of any fixed complete 22-complex. For higher dimensions k>2k>2, we get that p=O(n−1/k)p=O(n^{-1/k}) is an upper bound for the threshold probability of containing a subdivision of a fixed kk-dimensional complex.Comment: 15 page
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