Transforming Rectangles into Squares, with Applications to Strong Colorings

TL;DR

This paper proves that for any singular cardinal, there exists a function transforming rectangles into squares, linking strong colorings with classical partition relations, thus advancing understanding in set theory.

Contribution

It introduces a function that transforms rectangles into squares for singular cardinals and connects strong colorings with classical partition relations.

Findings

Existence of a rectangle-to-square transforming function for singular cardinals.

Equivalence between Shelah's strong coloring and negative partition relations.

Implications for the structure of colorings in set theory.

Abstract

It is proved that every singular cardinal $\lambda$ admits a function $RTS:[\lambda^+]^2\rightarrow[\lambda^+]^2$ that transforms rectangles into squares. Namely, for every cofinal subsets $A,B$ of $\lambda^+$, there exists a cofinal subset $C$ of $lambda^+$, such that $RTS[AxB]$ covers CxC. When combined with a recent result of Eisworth, this shows that Shelah's notion of strong coloring $Pr_1(\lambda^+,\lambda^+,\lambda^+,\cf(\lambda))$ coincides with the classical negative partition relation $\lambda^+\not\rightarrow[\lambda^+]^2_{\lambda^+}$.