# Partitioning the power set of $[n]$ into $C_k$-free parts

## TL;DR

This paper proves a sharp bound on partitioning the power set of [n] into triangle-free parts and explores a Ramsey-type problem for hypergraph colorings avoiding Berge-G subhypergraphs.

## Contribution

It establishes the exact minimum number of parts needed to guarantee a triangle in any partition and provides bounds for hypergraph colorings avoiding Berge-G configurations.

## Key findings

- Partition into 2^{n-2} triangle-free parts is optimal.
- Any partition into fewer than 2^{n-2} parts must contain a triangle.
- Upper bounds for hypergraph colorings avoiding Berge-G subhypergraphs.

## Abstract

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 (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 $G=S_3$.