Transforming Rectangles into Squares, with Applications to Strong Colorings

Abstract

It is proved that every singular cardinal $\lambda$ admits a function $RTS:[{\lambda }^{+}{]}^2\to [{\lambda }^{+}{]}^2$ that transforms rectangles into squares. Namely, for every cofinal subsets $A,B$ of ${\lambda }^{+}$, there exists a cofinal subset $C$ of $lambd{a}^{+}$, such that $RTS[AxB]$ covers CxC. When combined with a recent result of Eisworth, this shows that Shelah's notion of strong coloring $Pr_1(λ^+,λ^+,λ^+,\cf(λ))$ coincides with the classical negative partition relation ${\lambda }^{+}\nrightarrow [{\lambda }^{+}{]}_{{\lambda }^{+}}^2$.

preliminary version