Transforming Rectangles into Squares, with Applications to Strong Colorings

It is proved that every singular cardinal $λ$ admits a function $RTS:[λ^+]^2\rightarrow[λ^+]^2$ that transforms rectangles into squares. Namely, for every cofinal subsets $A,B$ of $λ^+$, 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 $Pr_1(λ^+,λ^+,λ^+,\cf(λ))$ coincides with the classical negative partition relation $λ^+\not\rightarrow[λ^+]^2_{λ^+}$.

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