Partitioning the power set of into -free parts
E Blaisdell, A Gyárfás, RA Krueger… - arXiv preprint arXiv …, 2018 - arxiv.org
arXiv preprint arXiv:1812.06448, 2018•arxiv.org
We show that for $ n\geq 3, n\ne 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, C\subseteq [n] $ such that $ A\cap B $, $ A\cap C $, and $
B\cap 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 …
subsets of $[n]=\{1, 2,\dots, n\} $, into $2^{n-2}-1$ parts, some part must contain a triangle---
three different subsets $ A, B, C\subseteq [n] $ such that $ A\cap B $, $ A\cap C $, and $
B\cap 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 …
We show that for , in any partition of , the set of all subsets of , into parts, some part must contain a triangle --- three different subsets such that , , and have distinct representatives. This is sharp, since by placing two complementary pairs of sets into each partition class, we have a partition into triangle-free parts. We also address a more general Ramsey-type problem: for a given graph , find (estimate) , the smallest number of colors needed for a coloring of , such that no color class contains a Berge- subhypergraph. We give an upper bound for for any connected graph which is asymptotically sharp (for fixed ) when , a cycle, path, or star with edges. Additional bounds are given for and .
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