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 $6^n$-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 $\infty$. Our model here is the medial regime.