Turan and Ramsey numbers in linear triple systems

TL;DR

This paper extends Turán and Ramsey number results from triangles to more complex triple configurations in linear triple systems, revealing new structural properties and unavoidable configurations in large hypergraphs.

Contribution

It generalizes Turán-type theorems to s-configurations and introduces extended s-configurations, establishing their Ramsey properties in Steiner triple systems.

Findings

Large linear triple systems contain s-configurations or extended s-configurations.

Most small unavoidable configurations are t-Ramsey, except possibly those with sail C_{15}.

The wicket D_4 is 1-Ramsey in all Steiner triple systems except the Fano plane.

Abstract

In this paper we study Tur\'an 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 Szemer\'edi is that for any fixed $c>0$ and large enough $n$ the following Tur\'an-type theorem holds. If a linear triple system on $n$ vertices has at least $cn^2$ edges then it contains a {\em triangle}: three pairwise intersecting triples without a common vertex. In this paper we extend this result from triangles to other triple systems, called {\em $s$-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 {\em extended $s$-configurations}. For these we cannot prove the corresponding Tur\'an-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 $C_{15}$ (configuration with blocks 123, 345, 561 and 147), are $t$-Ramsey for any $t\geq 1$. The most interesting one among them is the {\em wicket}, $D_4$, formed by three rows and two columns of a $3\times 3$ 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.