MathDB

Set theory from Schweitzer 2023

Source: Miklos Schweitzer 2023, Problem 1

March 16, 2024

set theory

Problem Statement

Prove that if

X{}

is an infinite set of cardinality

\kappa

then there is a collection

\mathcal{F}

of subsets of

X

such that [*]For any

A\subseteq X

with cardinality

\kappa

there exists

F\in\mathcal{F}

for which

A\cap F

has cardinality

\kappa,

and [*]

X

cannot be written as the union of less than

\kappa

sets from

\mathcal{F}

which all have cardinalities less than

\kappa.

Back to Problems View on AoPS