Partitioning the power set of \([n]\) into \(C_k\)-free parts

Eben Blaisdell¹ ; András Gyárfás² ; Robert A. Krueger³ ; Ronen Wdowinski⁴

¹Bucknell University
²Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences
³Miami University
⁴Rice University

The electronic journal of combinatorics, Tome 26 (2019) no. 3

Résumé

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 $A\cap B,A\cap C,B\cap C$ 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 when $G$ is a cycle, path, or star. Additional bounds are given when $G$ is a $4$-cycle and when $G$ is a claw.

Détail

DOI : 10.37236/8385

Classification : 05A18, 05D10, 05C65
Mots-clés : Ramsey-type problem