Georgia Institute of Technology College of Sciences

(This seminar has been rescheduled for April 17 (Thursday) 12-1pm. Generalized triangle T_r is an r-graph with edges {1,2,…,r}, {1,2,…,r-1, r+1} and {r,r+1, r+2, …,2r-2}. The family \Sigma_r consists of all r-graphs with three edges D_1, D_2, D_3 such that |D_1\cap D_2|=r-1 and D_1\triangle D_2\subset D_3. In 1989 it was conjectured by Frankl and Furedi that ex(n,T_r) = ex(n,\Sigma_r) for large enough n, where ex(n,F) is the Tur\'{a}n function. The conjecture was proven to be true for r=3, 4 by Frankl, Furedi and Pikhurko respectively. We settle the conjecture for r=5,6 and show that extremal graphs are blow-ups of the unique (11, 5, 4) and (12, 6, 5) Steiner systems. The proof is based on a technique for deriving exact results for the Tur\'{a}n function from “local stability" results, which has other applications. This is joint work with Sergey Norin.