Transforming Rectangles into Squares, with Applications to Strong Colorings

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It is proved that every singular cardinal λ admits a function RTS:[λ+]²→[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 Pr₁(λ+,λ+,λ+,(λ)) coincides with the classical negative partition relation λ+→[λ+]²λ₊.