Homogeneous Patterns in Ramsey Theory

Abstract

In this article, we investigate homogeneous versions of certain nonlinear Ramsey-theoretic results, with three significant applications. As the first application, we prove that for every finite coloring of ${\mathbb{Z}}^{+}$, there exist an infinite set $A$ and an arbitrarily large finite set $B$ such that $A\cup (A+B)\cup A\cdot B$ is monochromatic. This result resolves the finitary version of a question posed by Kra, Moreira, Richter, and Robertson regarding the partition regularity of $(A+B)\cup A\cdot B$ for infinite sets $A,B$ (see (Question 8.4, J. Amer. Math. Soc., 37 (2024))), which is closely related to a question of Erdős. As the second application, we make progress on a nonlinear extension of the partition regularity of Pythagorean triples. Specifically, we demonstrate that the equation $x^2+y^2=z^2+P(u_1,\dots ,u_n)$ is $2$-regular for certain appropriately chosen polynomials $P$ of any desired degree. Finally, as the third application, we establish a nonlinear variant of Rado's conjecture concerning the degree of regularity. We prove that for every $m,n\in {\mathbb{Z}}^{+}$, there exists an $m$-degree homogeneous equation that is $n$-regular but not $(n+1)$-regular. The case $m=1$ corresponds to Rado's conjecture, originally proven by Alexeev and Tsimerman (J. Combin. Theory Ser. A, 117 (2010), and later independently by Golowich (Electron. J. Combin. 21 (2014)).

Proofs are corrected. We welcome comments