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research
Maximal antichains of minimum size
Authors
Thomas Kalinowski
Uwe Leck
Ian T. Roberts
Publication date
1 January 2013
Publisher
View
on
arXiv
Abstract
Let
n
⩾
4
n\geqslant 4
n
⩾
4
be a natural number, and let
K
K
K
be a set
K
⊆
[
n
]
:
=
1
,
2
,
.
.
.
,
n
K\subseteq [n]:={1,2,...,n}
K
⊆
[
n
]
:=
1
,
2
,
...
,
n
. We study the problem to find the smallest possible size of a maximal family
A
\mathcal{A}
A
of subsets of
[
n
]
[n]
[
n
]
such that
A
\mathcal{A}
A
contains only sets whose size is in
K
K
K
, and
A
⊈
B
A\not\subseteq B
A
î€
⊆
B
for all
A
,
B
⊆
A
{A,B}\subseteq\mathcal{A}
A
,
B
⊆
A
, i.e.
A
\mathcal{A}
A
is an antichain. We present a general construction of such antichains for sets
K
K
K
containing 2, but not 1. If
3
∈
K
3\in K
3
∈
K
our construction asymptotically yields the smallest possible size of such a family, up to an
o
(
n
2
)
o(n^2)
o
(
n
2
)
error. We conjecture our construction to be asymptotically optimal also for
3
∉
K
3\not\in K
3
î€
∈
K
, and we prove a weaker bound for the case
K
=
2
,
4
K={2,4}
K
=
2
,
4
. Our asymptotic results are straightforward applications of the graph removal lemma to an equivalent reformulation of the problem in extremal graph theory which is interesting in its own right.Comment: fixed faulty argument in Section 2, added reference
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Last time updated on 10/05/2016