A simplicial complex is flag if all of its minimal non-faces have cardinality two. The celebrated upper bound theorem due to Stanley (1975) states that, in the class of all simplicial spheres, neighborly polytopes simultaneously maximize all the face numbers. On the other hand, for flag spheres, sharp upper bounds on the face numbers have only been established for spheres of dimension up to four. In addition, Lutz and Nevo conjectured that for flag 3-spheres with n vertices, the join of two circles of length as close as possible to n/2 is the unique maximizer of face numbers.

In this talk, I will prove the Lutz-Nevo conjecture and even extend it to the class of flag 3-manifolds. I will also show that the inequality part of the upper bound conjecture continues to hold for a much larger class of all flag Eulerian complexes and will characterize the cases of equality in this class.