Turan Number of the Generalized Triangle

Graph Theory Seminar
Thursday, April 17, 2014 - 12:05
1 hour (actually 50 minutes)
Skiles 005
McGill University (Montreal) and Georgia Tech
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.