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Miklós Schweitzer
2018 Miklós Schweitzer
2
Really neat sets
Really neat sets
Source: Miklós Schweitzer 2018 P2
November 10, 2018
college contests
Problem Statement
A family
F
\mathcal{F}
F
of sets is called really neat if for any
A
,
B
∈
F
A,B\in \mathcal{F}
A
,
B
∈
F
, there is a set
C
∈
F
C\in \mathcal{F}
C
∈
F
such that
A
∪
B
=
A
∪
C
=
B
∪
C
A\cup B = A\cup C=B\cup C
A
∪
B
=
A
∪
C
=
B
∪
C
. Let
f
(
n
)
=
min
{
max
A
∈
F
∣
A
∣
:
F
is really neat and
∣
∪
F
∣
=
n
}
.
f(n)=\min \left\{ \max_{A\in \mathcal{F}} |A| \colon \mathcal{F} \text{ is really neat and } |\cup \mathcal{F}| =n\right\} .
f
(
n
)
=
min
{
A
∈
F
max
∣
A
∣
:
F
is really neat and
∣
∪
F
∣
=
n
}
.
Prove that the sequence
f
(
n
)
/
n
f(n)/n
f
(
n
)
/
n
converges and find its limit.
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