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Part of 2023 Miklós Schweitzer

Problems(1)

Set theory from Schweitzer 2023

Source: Miklos Schweitzer 2023, Problem 1

3/16/2024
Prove that if XX{} is an infinite set of cardinality κ\kappa then there is a collection F\mathcal{F} of subsets of XX such that [*]For any A⊆XA\subseteq X with cardinality κ\kappa there exists F∈FF\in\mathcal{F} for which A∩FA\cap F has cardinality κ,\kappa, and [*]XX cannot be written as the union of less than κ\kappa sets from F\mathcal{F} which all have cardinalities less than κ.\kappa.
set theory