The coloring principle for the product of polish spaces and the Halpern and Läuchli's theorem

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 Läuchli’s theorem. The aim of this paper is to generaliza this consistency result to sets of colors of cardinality less than $2^{{\mathrm{\aleph }}_0}$. The proof presented here differs than the proof presented in arXiv:2209.04859.