Aristotelian poetry
Shimon Garti, Saharon Shelah
TL;DR
The paper investigates polarized relations with infinitely many colors and shows that one can force a model in which the positive relations $\binom{\omega_2}{\omega_1}\rightarrow \binom{n}{\omega_1}_\omega$ hold for all finite $n$ while the negative relation $\binom{\omega_2}{\omega_1}\nrightarrow \binom{\omega}{\omega_1}_\omega$ also holds, illustrating a reflection of Plato-Aristotle. The forcing construction $\mathbb{Q}$, assuming $MA$ and $2^\omega=\omega_2$, collapses cardinals below $\lambda$ to $\aleph_1$ and produces a witness coloring for the negative relation; positivity for finite $n$ follows from the $\omega_1$-Erdős nature of $\lambda$ via indiscernibles. The key contribution is reducing the consistency strength of the phenomenon to an $\omega_1$-Erdős cardinal and clarifying the limits of Baumgartner's stepping-up in this context. An independent, later result by Jing Zhang confirms the main theorem, reinforcing the identified threshold. The work also discusses deeper questions about measurable vs. weakly compact cardinals and the role of slim Kurepa trees in stationary-set versions of polarized relations.
Abstract
Jing Zhang proved the consistency of $\binom{ω_2}{ω_1}\rightarrow\binom{n}{ω_1}_ω$ for every $n\inω$ with the negative relation $\binom{ω_2}{ω_1}\nrightarrow\binomω{ω_1}_ω$. We reduce the consistency strength of this statement to an $ω_1$-Erdos cardinal.