Let $[n]=\{1,2,\ldots,n\}$ and $\mathscr{B}_n=\{A: A\subseteq [n]\}$. A family $\mathscr{A}\subseteq \mathscr{B}_n$ is a Sperner family if $A\nsubseteq B$ and $B\nsubseteq A$ for distinct $A,B\in\mathscr{A}$. Sperner's theorem states that the density of the largest Sperner family in $\mathscr{B}_n$ is $\binom{n}{\left\lceil{n/2}\right\rceil}/2^n$. The objective of this note is to show that the same holds if $\mathscr{B}_n$ is replaced by compressed ideals over $[n]$.
@article{10_37236_5750,
author = {Lili Mu and Yi Wang},
title = {A generalization of {Sperner's} theorem on compressed ideals},
journal = {The electronic journal of combinatorics},
year = {2016},
volume = {23},
number = {3},
doi = {10.37236/5750},
zbl = {1344.05145},
url = {http://geodesic.mathdoc.fr/articles/10.37236/5750/}
}
TY - JOUR
AU - Lili Mu
AU - Yi Wang
TI - A generalization of Sperner's theorem on compressed ideals
JO - The electronic journal of combinatorics
PY - 2016
VL - 23
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/5750/
DO - 10.37236/5750
ID - 10_37236_5750
ER -
%0 Journal Article
%A Lili Mu
%A Yi Wang
%T A generalization of Sperner's theorem on compressed ideals
%J The electronic journal of combinatorics
%D 2016
%V 23
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/5750/
%R 10.37236/5750
%F 10_37236_5750
Lili Mu; Yi Wang. A generalization of Sperner's theorem on compressed ideals. The electronic journal of combinatorics, Tome 23 (2016) no. 3. doi: 10.37236/5750
