Ramsey and Turán problem for sails
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.