The coloring principle for the product of polish spaces and the Halpern and L\"auchli's theorem

TL;DR

This paper extends a consistency result for a coloring principle on products of Polish spaces to larger sets of colors, providing a different proof and implications for Halpern and L"auchli's theorem.

Contribution

It generalizes previous results to uncountable color sets and offers a new proof approach for the coloring principle related to Polish space products.

Findings

Generalized the coloring principle to sets of colors smaller than continuum

Provided a new proof method differing from previous work

Connected the principle to Halpern and L"auchli's theorem

Abstract

In arXiv:2209.04859 Andy Zucker and Chris Lambie-Hanson proved the consistency result for some coloring principle for the products of polish spaces by at most countable many colors. This principle easy implies Halpern and L\"auchli's theorem. The aim of this paper is to generaliza this consistency result to sets of colors of cardinality less than $2^{\aleph_0}$. The proof presented here differs than the proof presented in arXiv:2209.04859.