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Jan 9, 2019 at 21:09 comment added Simon Rose Well, that kind of answers my next question as to whether or not simply connected might make a difference...
Jan 6, 2019 at 23:49 comment added Richard Stanley For every $n\geq d\geq 4$ there is a triangulation of a $(d-1)$-sphere (in fact, a simplicial polytope) with $n$ vertices, whose 1-skeleton is the complete graph $K_n$. These are the cyclic polytopes.
Jan 6, 2019 at 20:39 history answered Gjergji Zaimi CC BY-SA 4.0