Undecidability in the Ramsey theory of polynomial equations and Hilbert's tenth problem

TL;DR

This work connects Ramsey theory of polynomial equations to Hilbert's 10th problem by establishing undecidability and precise descriptive-set-theoretic complexity for partition and density regularity across countable integral domains. It develops a master-polynomial reduction framework that links $HTP(K)$ (and its variants) to partition-regular and density-regular properties, yielding $\Pi^0_2$-completeness in many cases and $\Sigma^0_1$-completeness for fixed-color variants. The authors introduce compactness and uniformity principles for density Ramsey theory on countable cancellative left amenable semigroups and extend natural extension results for measure-preserving semigroup actions. These results illuminate the computational boundaries of Ramsey-type questions for polynomial equations and offer a roadmap for further explorations of density regularity in broader algebraic and dynamical contexts, with several open problems highlighted. The work thereby advances our understanding of undecidability phenomena at the intersection of number theory, combinatorics, and descriptive set theory.

Abstract

We show that several sets of interest arising from the study of partition regularity and density Ramsey theory of polynomial equations over integral domains are undecidable. In particular, we show that the set of homogeneous polynomials $p \in \mathbb{Z}[x_1,\cdots,x_n]$ for which the equation $p(x_1,\cdots,x_n) = 0$ is partition regular over $\mathbb{Z}\setminus\{0\}$ is undecidable conditional on Hilbert's tenth problem for $\mathbb{Q}$. For other integral domains, we get the analogous result unconditionally. More generally, we determine the exact lightface complexity of the various sets of interest. For example, we show that the set of homogeneous polynomials $p \in \mathbb{F}_q(t)[x_1,\cdots,x_n]$ for which the equation $p(x_1,\cdots,x_n) = 0$ is partition regular over $\mathbb{F}_q(t)\setminus\{0\}$ is $Π_2^0$-complete. We also prove several other results of independent interest. These include a compactness principle and a uniformity principle for density Ramsey theory on countable cancellative left amenable semigroups, as well as the existence of the natural extension for measure preserving systems of countable cancellative left reversible semigroups.

Theorems & Definitions (70)

• Lemma 2.1
• Definition 2.2
• Theorem 2.3: TilingAmenableGroups
• Lemma 2.4
• proof
• Theorem 2.5
• Lemma 2.6
• proof
• Theorem 2.7: Pointwise ergodic theorem
• Theorem 2.8: Ergodic decomposition GeneralErgodicDecomposition