Connectivity of ample, conic and random simplicial complexes

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 6-conic complex is n-connected. This extends results by EvenZohar, 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.

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