# On topological minors in random simplicial complexes

Gundert A, Wagner U. 2016. On topological minors in random simplicial complexes. Proceedings of the American Mathematical Society. 144(4), 1815–1828.

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Author

Gundert, Anna;
Wagner, Uli

^{IST Austria}^{}Department

Abstract

For random graphs, the containment problem considers the probability that a binomial random graph 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/n. We consider a natural analogue of this question for higher-dimensional random complexes Xk(n, p), first studied by Cohen, Costa, Farber and Kappeler for k = 2. Improving previous results, we show that p = Θ(1/ √n) is the (coarse) threshold for containing a subdivision of any fixed complete 2-complex. For higher dimensions k > 2, we get that p = O(n−1/k) is an upper bound for the threshold probability of containing a subdivision of a fixed k-dimensional complex.

Publishing Year

Date Published

2016-04-01

Journal Title

Proceedings of the American Mathematical Society

Acknowledgement

This research was supported by the Swiss National Science Foundation (SNF Projects 200021-125309 and 200020-138230

Volume

144

Issue

4

Page

1815 - 1828

IST-REx-ID

### Cite this

Gundert A, Wagner U. On topological minors in random simplicial complexes.

*Proceedings of the American Mathematical Society*. 2016;144(4):1815-1828. doi:10.1090/proc/12824Gundert, A., & Wagner, U. (2016). On topological minors in random simplicial complexes.

*Proceedings of the American Mathematical Society*. American Mathematical Society. https://doi.org/10.1090/proc/12824Gundert, Anna, and Uli Wagner. “On Topological Minors in Random Simplicial Complexes.”

*Proceedings of the American Mathematical Society*. American Mathematical Society, 2016. https://doi.org/10.1090/proc/12824.A. Gundert and U. Wagner, “On topological minors in random simplicial complexes,”

*Proceedings of the American Mathematical Society*, vol. 144, no. 4. American Mathematical Society, pp. 1815–1828, 2016.Gundert, Anna, and Uli Wagner. “On Topological Minors in Random Simplicial Complexes.”

*Proceedings of the American Mathematical Society*, vol. 144, no. 4, American Mathematical Society, 2016, pp. 1815–28, doi:10.1090/proc/12824.**All files available under the following license(s):**

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