An uncountable version of Pt\'ak's combinatorial lemma

TL;DR

This paper investigates an uncountable extension of Pták's combinatorial lemma, demonstrating its independence from ZFC for  and offering conditions for larger cardinals.

Contribution

It establishes the independence of the uncountable Pták's lemma from ZFC and provides sufficient conditions for its validity at larger cardinals.

Findings

Independence of the lemma from ZFC at 

Sufficient conditions for validity at larger cardinals

Insights into combinatorial set theory and large cardinals

Abstract

In this note we are concerned with the validity of an uncountable analogue of a combinatorial lemma due to Vlastimil Pt\'ak. We show that the validity of the result for $\omega_1$ can not be decided in ZFC alone. We also provide a sufficient condition, for a class of larger cardinals.