### Abstract

We show that for n ≥ 3, n ≠ 5, in any partition of P(n), the set of all subsets of [n] = {1, 2, …, n}, into 2^{n−2} − 1 parts, some part must contain a triangle — three different subsets A, B, C ⊆ [n] such that A ∩ B, A ∩ C, B ∩ 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 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 when G is a cycle, path, or star. Additional bounds are given when G is a 4-cycle and when G is a claw. X_{1} = {[5], [4]}, Y_{1} = {24, 124, 234, 245}, Y_{2} = {13, 123, 134, 135}, Z_{1} = {12, 35, 1235, 345}. In fact, (1) is the only partition of P^{∗}(5) into at most seven C_{3}-free parts (up to permutations), implying f(5, C_{3}) = 7. (Note that property 4 of Lemma 7 is not true for n = 5, as witnessed by the ‘crowns’ Y_{1} and Y_{2} in (1).).

Original language | English |
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Article number | P3.38 |

Journal | Electronic Journal of Combinatorics |

Volume | 26 |

Issue number | 3 |

Publication status | Published - Jan 1 2019 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

_{k}-free parts.

*Electronic Journal of Combinatorics*,

*26*(3), [P3.38].