Turán and Ramsey numbers in linear triple systems

Abstract

In this paper we study Turán and Ramsey numbers in linear triple systems, defined as 3-uniform hypergraphs in which any two triples intersect in at most one vertex.A famous result of Ruzsa and Szemerédi is that for any fixed c>0 and large enough n the following Turán-type theorem holds. If a linear triple system on n vertices has at least cn2 edges then it contains a triangle: three pairwise intersecting triples without a common vertex. In this paper we extend this result from triangles to other triple systems, called s-configurations. The main tool is a generalization of the induced matching lemma from aba-patterns to more general ones.We slightly generalize s-configurations to extendeds-configurations. For these we cannot prove the corresponding Turán-type theorem, but we prove that they have the weaker, Ramsey property: they can be found in any t-coloring of the blocks of any sufficiently large Steiner triple system. Using this, we show that all unavoidable configurations with at most 5 blocks, except possibly the ones containing the sail C15 (configuration with blocks 123, 345, 561 and 147), are t-Ramsey for any t≥1. The most interesting one among them is the wicket, D4, formed by three rows and two columns of a 3 × 3 point matrix. In fact, the wicket is 1-Ramsey in a very strong sense: all Steiner triple systems except the Fano plane must contain a wicket.