For a given positive integer $k$ we say that a family of subsets of $[n]$ is $k$-antichain saturated if it does not contain $k$ pairwise incomparable sets, but whenever we add to it a new set, we do find $k$ such sets. The size of the smallest such family is denoted by $\text{sat}^*(n, \mathcal A_{k})$. Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan conjectured that $\text{sat}^*(n, \mathcal A_{k})=(k-1)n(1+o(1))$, and proved this for $k\leq4$. In this paper we prove this conjecture for $k=5$ and $k=6$. Moreover, we give the exact value for $\text{sat}^*(n, \mathcal A_5)$ and $\text{sat}^*(n, \mathcal A_6)$. We also give some open problems inspired by our analysis.
@article{10_37236_11262,
author = {Irina {\DH}ankovi\'c and Maria-Romina Ivan},
title = {Saturation for small antichains},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {1},
doi = {10.37236/11262},
zbl = {1539.06003},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11262/}
}
TY - JOUR
AU - Irina Ðanković
AU - Maria-Romina Ivan
TI - Saturation for small antichains
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/11262/
DO - 10.37236/11262
ID - 10_37236_11262
ER -
%0 Journal Article
%A Irina Ðanković
%A Maria-Romina Ivan
%T Saturation for small antichains
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/11262/
%R 10.37236/11262
%F 10_37236_11262
Irina Ðanković; Maria-Romina Ivan. Saturation for small antichains. The electronic journal of combinatorics, Tome 30 (2023) no. 1. doi: 10.37236/11262
