Turán and Ramsey numbers in linear triple systems

Abstract In this paper we study Turan and Ramsey numbers in linear triple systems, defined as 3-uniform hypergraphs which any two triples intersect at most one vertex. A famous result of Ruzsa Szemeredi is that for fixed c > 0 large enough n the following Turan-type theorem holds. If a system on vertices has least 2 edges then it contains triangle: three pairwise intersecting without common extend from triangles to other called s -configurations. The main tool generalization induced matching lemma b -patterns more general ones. We slightly generalize -configurations extended For these cannot prove corresponding theorem, but they have weaker, property: can be found t -coloring blocks sufficiently Steiner system. Using this, show all unavoidable configurations with 5 blocks, except possibly ones containing sail C 15 (configuration 123, 345, 561 147), are -Ramsey ? 1 . interesting among them wicket, D 4 , formed by rows columns 3 × 3 point matrix. fact, wicket 1-Ramsey very strong sense: systems Fano plane must contain wicket.