Transforming Rectangles into Squares, with Applications to Strong Colorings

2011/03/15 by Assaf Rinot, Rinot, Assaf
Mathematics · #03E02 #FOS: Mathematics #Logic (math.LO) #math.LO #msc:03E02

paper · pdf · doi:10.48550/arxiv.1103.2838

preliminary version

arxiv created 2011/03/15 · arxiv updated 2011/03/16

Abstract

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⁺,\cf(λ)) coincides with the classical negative partition relation λ⁺\not→[λ⁺]²_λ⁺.

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