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Romanian Masters of Mathematics Collection
2017 Romanian Master of Mathematics
3
3
Part of
2017 Romanian Master of Mathematics
Problems
(1)
Proper Subset
Source: Romanian Masters 2017 D1 P3
2/25/2017
Let
n
n
n
be an integer greater than
1
1
1
and let
X
X
X
be an
n
n
n
-element set. A non-empty collection of subsets
A
1
,
.
.
.
,
A
k
A_1, ..., A_k
A
1
,
...
,
A
k
of
X
X
X
is tight if the union
A
1
∪
⋯
∪
A
k
A_1 \cup \cdots \cup A_k
A
1
∪
⋯
∪
A
k
is a proper subset of
X
X
X
and no element of
X
X
X
lies in exactly one of the
A
i
A_i
A
i
s. Find the largest cardinality of a collection of proper non-empty subsets of
X
X
X
, no non-empty subcollection of which is tight.Note. A subset
A
A
A
of
X
X
X
is proper if
A
≠
X
A\neq X
A
=
X
. The sets in a collection are assumed to be distinct. The whole collection is assumed to be a subcollection.
RMM
Set systems
RMM 2017