Random complexes with free involution

We introduce a new model for random simplicial complexes which with high probability generates a complex that has a simply-connected double cover. Hence we develop a model for random simplicial complexes with fundamental group $\mathbb{Z}\mathrm{/}2\mathbb{Z}$. We establish results about the typical asymptotic topology of these complexes. As a consequence we give bounds for the dimension $d$ such that $\mathbb{Z}\mathrm{/}2\mathbb{Z}$-equivariant maps from the double cover to ${\mathbb{R}}^d$ have zeros with high probability, thus establishing a random Borsuk–Ulam theorem. We apply this to derive a structural result for pairs of non-adjacent cliques in Erdős–Rényi random graphs.