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Miklós Schweitzer
2023 Miklós Schweitzer
7
7
Part of
2023 Miklós Schweitzer
Problems
(1)
Sequence of sets of size 4
Source: Miklos Schweitzer 2023, Problem 7
11/7/2023
Prove that there exist two subsets
D
,
K
D, K
D
,
K
of
N
2
\mathbb{N}^2
N
2
such that for any
4
4
4
-element sets
A
1
,
A
2
A_1, A_2
A
1
,
A
2
we have
∣
A
1
∩
A
2
∣
=
1
|A_1 \cap A_2|=1
∣
A
1
∩
A
2
∣
=
1
if and only if there exist
4
4
4
-element sets
A
3
,
A
4
,
…
A_3, A_4, \ldots
A
3
,
A
4
,
…
, such that
A
i
∩
A
j
=
∅
A_i \cap A_j=\emptyset
A
i
∩
A
j
=
∅
for all
(
i
,
j
)
∈
D
(i, j) \in D
(
i
,
j
)
∈
D
and
∣
A
i
∩
A
j
∣
=
2
|A_i \cap A_j|=2
∣
A
i
∩
A
j
∣
=
2
for all
(
i
,
j
)
∈
K
(i, j) \in K
(
i
,
j
)
∈
K
.