The upper bound theorem asserts that among all *d*-dimensional polytopes with *n* vertices, the cyclic polytope maximizes the number of facets (and *k*-faces for every *k*). It was proved by McMullen for polytopes in 1970, and by Stanley for general triangulations of spheres in 1975. This theorem is related to a lot of mathematics (and also computational geometry) and there are many interesting extensions and related conjectures.

Richard Stanley posted (on his homepage which is full with interesting things) an article describing *How the proof of the upper bound theorem for triangulations of spheres was found*.

It is interesting how, for Richard, the work oמ face-numbers of polytopes started with his work on integer points in polytopes and especially the Anand-Dumir-Gupta” conjecture on enumeration of “magic squares.” (See this survey article by Winfried Bruns.) Integer points in polytopes, and face numbers represent the two initial chapters of Richard’s “green book” on Commutative algebra and combinatorics. Both these topics are related to commutative algebra and to the algebraic geometry of toric varieties.

See also these relevant posts “(Eran Nevo) The g-conjecture II: The commutative algebra connection,), (Eran Nevo) The g-conjecture I, and How the g-conjecture came about. Various results and problems related to the upper bound theorem can be found in Section 2 of my paper *Combinatorics with a Geometric Flavor*.