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research
Minimizing the regularity of maximal regular antichains of 2- and 3-sets
Authors
Thomas Kalinowski
Uwe Leck
Christian Reiher
Ian T. Roberts
Publication date
1 January 2014
Publisher
View
on
arXiv
Abstract
Let
n
β©Ύ
3
n\geqslant 3
n
β©Ύ
3
be a natural number. We study the problem to find the smallest
r
r
r
such that there is a family
A
\mathcal{A}
A
of 2-subsets and 3-subsets of
[
n
]
=
{
1
,
2
,
.
.
.
,
n
}
[n]=\{1,2,...,n\}
[
n
]
=
{
1
,
2
,
...
,
n
}
with the following properties: (1)
A
\mathcal{A}
A
is an antichain, i.e. no member of
A
\mathcal A
A
is a subset of any other member of
A
\mathcal A
A
, (2)
A
\mathcal A
A
is maximal, i.e. for every
X
β
2
[
n
]
β
A
X\in 2^{[n]}\setminus\mathcal A
X
β
2
[
n
]
β
A
there is an
A
β
A
A\in\mathcal A
A
β
A
with
X
β
A
X\subseteq A
X
β
A
or
A
β
X
A\subseteq X
A
β
X
, and (3)
A
\mathcal A
A
is
r
r
r
-regular, i.e. every point
x
β
[
n
]
x\in[n]
x
β
[
n
]
is contained in exactly
r
r
r
members of
A
\mathcal A
A
. We prove lower bounds on
r
r
r
, and we describe constructions for regular maximal antichains with small regularity.Comment: 7 pages, updated reference
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Last time updated on 30/10/2017