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In this paper we study Turan 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 Szemeredi is that for any fixed $c>0$ and large enough $n$ the following Turan-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 Turan-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 $tgeq 1$. The most interesting one among them is the {em wicket}, $D_4$, formed by three rows and two columns of a $3times 3$ 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.
We call a $4$-cycle in $K_{n_{1}, n_{2}, n_{3}}$ multipartite, denoted by $C_{4}^{text{multi}}$, if it contains at least one vertex in each part of $K_{n_{1}, n_{2}, n_{3}}$. The Turan number $text{ex}(K_{n_{1},n_{2},n_{3}}, C_{4}^{text{multi}})$ $bi
A Berge-$K_4$ in a triple system is a configuration with four vertices $v_1,v_2,v_3,v_4$ and six distinct triples ${e_{ij}: 1le i< j le 4}$ such that ${v_i,v_j}subset e_{ij}$ for every $1le i<jle 4$. We denote by $cal{B}$ the set of Berge-$K_4$ confi
A {em special four-cycle } $F$ in a triple system consists of four triples {em inducing } a $C_4$. This means that $F$ has four special vertices $v_1,v_2,v_3,v_4$ and four triples in the form $w_iv_iv_{i+1}$ (indices are understood $pmod 4$) where th
For given graphs $G$ and $F$, the Turan number $ex(G,F)$ is defined to be the maximum number of edges in an $F$-free subgraph of $G$. Foucaud, Krivelevich and Perarnau and later independently Briggs and Cox introduced a dual version of this problem w
In this paper, we consider a variant of Ramsey numbers which we call complementary Ramsey numbers $bar{R}(m,t,s)$. We first establish their connections to pairs of Ramsey $(s,t)$-graphs. Using the classification of Ramsey $(s,t)$-graphs for small $s,