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تسجيل مستخدم جديدPartitioning the power set of $[n]$ into $C_k$-free parts
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We show that for $n geq 3, n e 5$, in any partition of $mathcal{P}(n)$, the set of all subsets of $[n]={1,2,dots,n}$, into $2^{n-2}-1$ parts, some part must contain a triangle --- three different subsets $A,B,Csubseteq [n]$ such that $Acap B$, $Acap C$, and $Bcap C$ have distinct representatives. This is sharp, since by placing two complementary pairs of sets into each partition class, we have a partition into $2^{n-2}$ triangle-free parts. We also address a more general Ramsey-type problem: for a given graph $G$, find (estimate) $f(n,G)$, the smallest number of colors needed for a coloring of $mathcal{P}(n)$, such that no color class contains a Berge-$G$ subhypergraph. We give an upper bound for $f(n,G)$ for any connected graph $G$ which is asymptotically sharp (for fixed $k$) when $G=C_k, P_k, S_k$, a cycle, path, or star with $k$ edges. Additional bounds are given for $G=C_4$ and $G=S_3$.
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