Transforming Rectangles into Squares, with Applications to Strong Colorings

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 $P{r}_1({\lambda }^{+},{\lambda }^{+},{\lambda }^{+},\text{cf}(\lambda ))$ coincides with the classical negative partition relation ${\lambda }^{+}\to ̸[{\lambda }^{+}{]}_{{\lambda }^{+}}^2$.