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Let Δ be a triangulation of a ( d − 1)‐dimensional sphere with n vertices. The Upper Bound Conjecture states that the number of i ‐dimensional faces of Δ is less than or equal to a certain explicit number c i ( n, d ). A proof is given of a more general result. The proof uses the result, proved by G. Reisner, that a certain commutative ring associated with Δ is a Cohen‐Macaulay ring.
Richard P. Stanley (Sun,) studied this question.