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This is the second of two papers investigating for which positive integers $m$ there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$ (the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion). In the first part, the sizes of
Externí odkaz:
http://arxiv.org/abs/2302.07062
This is the second in a sequence of three papers investigating the question for which positive integers $m$ there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$ (the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion). In
Externí odkaz:
http://arxiv.org/abs/2106.02230
Publikováno v:
Order, Volume 40, pages 537-574, (2023)
Extending a classical theorem of Sperner, we characterize the integers $m$ such that there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$, that is, the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion. As an important in
Externí odkaz:
http://arxiv.org/abs/2106.02226
Publikováno v:
Order 38, 441-453 (2021)
Building on classical theorems of Sperner and Kruskal-Katona, we investigate antichains $\mathcal F$ in the Boolean lattice $B_n$ of all subsets of $[n]:=\{1,2,\dots,n\}$, where $\mathcal F$ is flat, meaning that it contains sets of at most two conse
Externí odkaz:
http://arxiv.org/abs/1704.00067
Akademický článek
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Autor:
Roberts, Ian T.
This thesis considers three related structures on finite sets and outstanding conjectures on two of them. Several new problems and conjectures are stated.A union-closed collection of sets is a collection of sets which contains the union of each pair
Publikováno v:
Australasian Journal of Combinatorics 64 (2016), 277-288
Let $n\geqslant 3$ be a natural number. We study the problem to find the smallest $r$ such that there is a family $\mathcal{A}$ of 2-subsets and 3-subsets of $[n]=\{1,2,...,n\}$ with the following properties: (1) $\mathcal{A}$ is an antichain, i.e. n
Externí odkaz:
http://arxiv.org/abs/1206.3752
Publikováno v:
The Electronic Journal of Combinatorics 20(1) (2013) #P3
Let $n\geqslant 4$ be a natural number, and let $K$ be a set $K\subseteq [n]:={1,2,...,n}$. We study the problem to find the smallest possible size of a maximal family $\mathcal{A}$ of subsets of $[n]$ such that $\mathcal{A}$ contains only sets whose
Externí odkaz:
http://arxiv.org/abs/1206.3007
Publikováno v:
Order; Oct2023, Vol. 40 Issue 3, p537-574, 38p
Publikováno v:
In Discrete Applied Mathematics 30 January 2014 163 Part 2:165-180