Transforming Rectangles into Squares, with Applications to Strong Colorings

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 $\lambda^+\not\rightarrow[\lambda^+]^2_{\lambda^+}$.