# Title: Simplicial face numbers via extremal graph theory

## ABSTRACT

Seminar is on Thursday in Padelford C-401 at 10:30-11:30pm.
There is no pre-seminar.

One aspect of topological combinatorics is face enumeration, and one of its main problems is to understand how the topology of a space affects the face numbers of its simplicial triangulations. For example, Euler proved that any triangulation of $S^2$ satisfies $(f_0,f_1,f_2)=(n,3n-6,2n-4)$, where $f_0,f_1,f_2$ are the number of vertices, edges and triangles, respectively. Since then various algebraic and topological tools have been developed to study face numbers of spheres and manifolds.

We are interested in this type of question for the natural class of flag complexes, which are just clique complexes of graphs. The good news is that such a complex is completely determined by its 1-skeleton but the bad news is that clique numbers of graphs are not understood nearly as well as face numbers of arbitrary complexes. The structure of sparsest flag triangulations of spheres (a lower bound type of statement) is mostly conjectural and related to the Charney-Davis conjecture and its $\gamma$-vector generalizations by Gal. In this talk I will concentrate on the densest flag triangulations (an upper bound statement) of manifolds. I will introduce a method that allows to determine the flag triangulations with maximal (or close to maximal) face numbers and can be tailored to spheres, (homology) manifolds and some classes of pseudomanifolds. It has two ingredients: first, we use tools from extremal graph theory to get a rough structure of the maximizer, and then we rigidify it using whatever topological properties we have at hand.

Joint work with Jan Hladky (Prague).