Connectivity of Ample, Conic, and Random Simplicial Complexes

Abstract

A simplicial complex is r-conic if every subcomplex of at most r vertices is contained in the star of a vertex. A 4-conic complex is simply connected. We prove that an 8-conic complex is 2-connected. In general a (2n+1)-conic complex need not be n-connected but a 5n-conic complex is n-connected. This extends results by Even-Zohar, Farber, and Mead on ample complexes and answers two questions raised in their paper. Our results together with theirs imply that the probability of a complex being n-connected tends to 1 as the number of vertices tends to ∞⁠. Our model here is the medial regime.