Afshin Goodarzi: Convex Hull of Face Vectors of Colored Complexes

The Turan graph T(n,r) is the complete r-partite graph of order n with cardinality of the maximal independent sets "as equal as possible". 

In 1941 Turan proved that among all graphs of order $n$ and clique number less than r+1, the Turan graph T(n,r) has the maximum number of edges. This result is a cornerstone of Extremal Graph Theory. 
Later, in 1949, Zykov generalized Turan's result by showing that T(n,r) has the maximum number of k-cliques, for any k. 

In 1997, Dmitry Kozlov conjectured that the convex hull of face vectors of r-colorable complexes on n vertices is generated by the face vectors of skeleta of the clique complexes of the Turan graph T(n,r).
 
In this talk we verify Kozlov's conjecture. As part of the proof we derive a generalization of Zykov's theorem.