Let $\mathcal{B}(n)$ denote the collection of all set partitions of $[n]$. Suppose $\mathcal A_1,\mathcal A_2\subseteq \mathcal{B}(n)$ are cross-intersecting i.e. for all $A_1\in \mathcal A_1$ and $A_2\in \mathcal A_2$, we have $A_1\cap A_2\neq\varnothing$. It is proved that for sufficiently large $n$,\[ \vert \mathcal A_1\vert\vert \mathcal A_2\vert\leq B_{n-1}^2\]where $B_{n}$ is the $n$-th Bell number. Moreover, equality holds if and only if $\mathcal{A}_1=\mathcal A_2$ and $\mathcal A_1$ consists of all set partitions with a fixed singleton.
@article{10_37236_2191,
author = {Cheng Yeaw Ku and Kok Bin Wong},
title = {On cross-intersecting families of set partitions},
journal = {The electronic journal of combinatorics},
year = {2012},
volume = {19},
number = {4},
doi = {10.37236/2191},
zbl = {1267.05035},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2191/}
}
TY - JOUR
AU - Cheng Yeaw Ku
AU - Kok Bin Wong
TI - On cross-intersecting families of set partitions
JO - The electronic journal of combinatorics
PY - 2012
VL - 19
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/2191/
DO - 10.37236/2191
ID - 10_37236_2191
ER -
%0 Journal Article
%A Cheng Yeaw Ku
%A Kok Bin Wong
%T On cross-intersecting families of set partitions
%J The electronic journal of combinatorics
%D 2012
%V 19
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/2191/
%R 10.37236/2191
%F 10_37236_2191
Cheng Yeaw Ku; Kok Bin Wong. On cross-intersecting families of set partitions. The electronic journal of combinatorics, Tome 19 (2012) no. 4. doi: 10.37236/2191
