f-vectors of flag complexes

Jan Hladký

Abstract

An f-vector is an enumeration of faces of all dimensions of a given geometric object. So, for example the f-vector of a 3-cube is (8,12,6) as it has 8 vertices, 12 edges, and 6 sides. f-vectors and its relatives (called gamma- and h-vectors) are central objects of study in geometric combinatorics. What f vectors can be attained by a given class of geometric objects? Or, can we at least get upper bounds for face numbers of a given dimension ("upper-bound theorems"). With Michal Adamaszek (Copenhagen) we applied methods from extremal graph theory and got an optimal upper-bound theorem for flag triangulations of manifolds of odd dimension. I will explain the connection. Only basic graph theory (Euler's formula) will be assumed, otherwise the talk will be self-contained.