Abstract
Given two combinatorial notions P0 and P1, can we encode P0 via P1? In this paper we address the question where P0 is 3-coloring of integers and P1 is a product of finitely many 2-colorings of integers.
We firstly reduce the question to a lemma which asserts that certain Π 01 -classes of colorings admit two members violating a particular combinatorial constraint. Then we digress to see how complex the class must be to maintain the constraint.
We weaken the two members in the lemma in a certain way to address a question of Cholak, Dzhafarov, Hirschfeldt and Patey, concerning a sort of Weihrauch degree of stable Ramsey’s theorem for pairs. It turns out the resulted strengthening of the lemma is a basis theorem for a Π 01 -class with additional constraints. We look at several such variants of basis theorem, among ahich some are unknown. We end up by introducing some results and questions concerning a product of infinitely many colorings.
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Liu, L. The coding power of a product of partitions. Isr. J. Math. 255, 645–683 (2023). https://doi.org/10.1007/s11856-023-2475-y
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DOI: https://doi.org/10.1007/s11856-023-2475-y