|Discrete Mathematics and Optimization Seminar
Monday December 1st at 4.30pm
Title. The exact Turán number of the Fano plane.
The Fano plane is the unique hypergraph with 7 triples on 7 vertices in which
every pair of vertices is contained in a unique triple. Its edges correspond to
the lines of the projective plane over the field with two elements. The Turán
problem is to find the maximum numebr of edges in a 3-uniform hypergraph
on n vertices not containing a Fano plane.
Noting that the Fano plane is not 2-colourable, but becomes so if one deletes an
edge, a natural candidate is the largest 2-colourable 3-uniform hypergraph
on n vertices. This is obtained by partitioning the vertices into two parts, of sizes
differing by at most one, and taking all the triples which intersect both
of them. Denote this hypergraph by H2(n).
We show that for sufficiently large n, the unique largest 3-uniform hypergraph
on n vertices not containing a Fano plane is H2(n), thus proving a conjecture
of V. Sós raised in 1976. This is joint work with Benny Sudakov.