Canonization of analytic equivalences on the Carlson-Simpson forcing

TL;DR

This paper establishes a canonization theorem for the Carlson-Simpson forcing, extending previous results to uncountable colorings with analytic equivalence relations, advancing the understanding of partition regularities.

Contribution

It generalizes the weak Carlson-Simpson theorem to uncountable colorings with analytic equivalence relations, providing a broader canonization framework.

Findings

Proves a canonization result for Carlson-Simpson forcing.

Extends the weak Carlson-Simpson theorem to uncountable colorings.

Shows that analytic equivalence relations can be canonized in this setting.

Abstract

We prove a canonization result for the Carlson-Simpson forcing in the spirit of \cite{KSZ}. We generalize the weak form of the Carlson-Simpson theorem (\cite{CaSi}) dealing with partitions without free blocks: instead of dealing with finite Borel (resp. Baire-property) colorings we deal with (uncountable) colorings such that the corresponding equivalence relation (two partitions are equivalent if they are colored by the same color) is analytic.