Partitioning the power set of [n] into ck-free parts

Eben Blaisdell, Robert A. Krueger, András Gyárfás, Ronen Wdowinski

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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 2n−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 2n−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. X1 = {[5], [4]}, Y1 = {24, 124, 234, 245}, Y2 = {13, 123, 134, 135}, Z1 = {12, 35, 1235, 345}. In fact, (1) is the only partition of P(5) into at most seven C3-free parts (up to permutations), implying f(5, C3) = 7. (Note that property 4 of Lemma 7 is not true for n = 5, as witnessed by the ‘crowns’ Y1 and Y2 in (1).).

Original languageEnglish
Article numberP3.38
JournalElectronic Journal of Combinatorics
Volume26
Issue number3
Publication statusPublished - 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

Blaisdell, E., Krueger, R. A., Gyárfás, A., & Wdowinski, R. (2019). Partitioning the power set of [n] into ck-free parts. Electronic Journal of Combinatorics, 26(3), [P3.38].