Michał Adamaszek: Dense flag triangulations of 3-spheres and extremal graph theory

In his work introducing the gamma-vector of flag spheres Gal gives an almost complete classification of face vectors of flag simplicial 3-spheres (or, more generally, homology 3-spheres). The missing ingredient is to show that a sufficiently dense flag triangulation of the 3-sphere must be a join of two circles. We prove this fact for spheres with sufficiently many vertices. The proof uses techinques of extremal graph theory. I will also speculate about possible other applications of our approach. Joint work with Jan Hladky.