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تسجيل مستخدم جديدThe Turan number of Berge-K_4 in triple systems
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Andras Gyarfas
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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-$K_4$ configurations. A triple system is $cal{B}$-free if it does not contain any member of $cal{B}$. We prove that the maximum number of triples in a $cal{B}$-free triple system on $nge 6$ points is obtained by the balanced complete $3$-partite triple system: all triples ${abc: ain A, bin B, cin C}$ where $A,B,C$ is a partition of $n$ points with $$leftlfloor{nover 3}rightrfloor=|A|le |B|le |C|=leftlceil{nover 3}rightrceil.$$
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some hyperedge. For a graph $G=(V,E)$, a hypergraph $mathcal{H}$ is called a textit{Berge}-$G$, denoted by $BG$, if there exists an injection $f: E(G) to E(mathcal{H})$ such that for every $e in E(G)$, $e subseteq f(e)$. In this note, we define a new type of Ramsey number, namely the emph{cover Ramsey number}, denoted as $hat{R}^R(BG_1, BG_2)$, as the smallest integer $n_0$ such that for every covering $R$-uniform hypergraph $mathcal{H}$ on $n geq n_0$ vertices and every $2$-edge-coloring (blue and red) of $mathcal{H}$ , there is either a blue Berge-$G_1$ or a red Berge-$G_2$ subhypergraph. We show that for every $kgeq 2$, there exists some $c_k$ such that for any finite graphs $G_1$ and $G_2$, $R(G_1, G_2) leq hat{R}^{[k]}(BG_1, BG_2) leq c_k cdot R(G_1, G_2)^3$. Moreover, we show that for each positive integer $d$ and $k$, there exists a constant $c = c(d,k)$ such that if $G$ is a graph on $n$ vertices with maximum degree at most $d$, then $hat{R}^{[k]}(BG,BG) leq cn$.
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