We consider simplicial polytopes, and more general simplicial complexes, without missing faces above a fixed dimension.
(F is a missing face of a complex K if the boundary of F is contained in K and F is not in K.)
We conjecture sharp analogues of McMullen's generalized lower bounds, and of Barnette's lower bounds, for these families of complexes. This gives a hierarchy of conjectures on lower bounds on face numbers, interpolating between McMullen's generalized lower bound conjecture for simplicial spheres and Gal's conjecture for flag spheres. We provide partial results on these conjectures.
Time permitting, a conjecture and results joint with Kyle Petersen on upper bounds on face numbers of flag spheres will be presented.
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