Hungarian Cubes

TL;DR

The paper investigates positive cube/polarized relations at infinite cardinals, focusing on the strong cube relation $(\nu\mu\lambda)\rightarrow(\nu\mu\lambda)$. It analyzes terraced-cube phenomena, proving negative results when $2^\lambda\le\lambda^{++}$ and developing a generalized terraced framework under tailored pcf-structural assumptions to obtain positive relations. Central to the results is a forcing construction starting from two supercompact cardinals that arranges $u_\mu<2^\mu$ and then forces a singular strong-limit $\lambda$ with controlled tcf targets, so that the positive relation survives to the final model. The main theorem shows the consistency of $(\nu\mu\lambda)\rightarrow(\nu\mu\lambda)$ in the regime $\lambda<\mu={\rm cf}(\mu)<\nu={\rm cf}(\nu)=2^\mu$, illustrating how large-cardinal assumptions and pcf techniques can yield robust partition relations at infinite cardinals. These results bridge partition calculus, forcing, and pcf theory, and suggest avenues for extending positive cube relations to broader configurations.

Abstract

We prove a positive polarized cube relation for infinite cardinals.

Theorems & Definitions (9)

• Proposition 1.1
• Claim 1.2
• Theorem 1.3
• Theorem 2.1
• Theorem 2.3
• Remark 2.4
• Claim 3.1
• Lemma 3.2
• Theorem 3.3