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Compactness property with partitions of N

Source: KoMaL A. 836

December 13, 2022

combinatoricstopologykomal

Problem Statement

For every

i \in \mathbb{N}

let

A_i

,

B_i

and

C_i

be three finite and pairwise disjoint subsets of

\mathbb{N}

. Suppose that for every pairwise disjoint sets

A

,

B

and

C

with union

\mathbb N

there exists

i\in \mathbb{N}

such that

A_i \subset A

,

B_i \subset B

and

C_i \subset C

. Prove that there also exists a finite

S\subset \mathbb{N}

such that for every pairwise disjoint sets

A

,

B_i \subset B

and

C

with union

\mathbb N

there exists

i\in S

such that

A_i \subset A

,

B_i \subset B

and

C_i \subset C

.
Submitted by András Imolay, Budapest

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