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Ramsey and Turán problem for sails

The CEU Campus
Wednesday, February 7, 2018, 2:00 pm

Abstract:  A triple system is linear if any two triples in it intersect in at most one point. There are 16 non-isomorphic linear triple systems, perhaphs  the Pasch configuration is the most famous among them, obtained from the Fano plane by deleting one point and the three triples containing it.  I introduce  another member of the group, the sail: three triples through a point P and a fourth triple meeting the other three in points different from P. In  an undergraduate research course during the summer of 2016 at BSM we tried to find which configurations C among the 16 have the Ramsey property:  for sufficiently large admissible $n$ every 2-coloring of the triples of any Steiner triple system with n points, there is a monochromatic copy of C.  We could decide this apart from one configuration: the sail.   Then, in 2017 with Füredi we looked at the "Turán side of the coin",  asking about the largest number of triples in a  linear triple system on n points that does not contain sails.  The answer is $n^2/9$  with equality if and only if n is divisible by three and we have a transversal design on three groups. However, for $n=3k+1$ the problem remains open.