Transforming rectangles into squares, with applications to strong colorings

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⊆λ+, such that rts[A⊛B]⊇C⊛C.As a corollary, we get that for every uncountable cardinal λ, the classical negative partition relation λ+⁄→[λ+]λ+2 coincides with the following syntactically stronger statement. There exists a function f:[λ+]2→λ+ such that for every positive integer n, every family A⊆[λ+]n of size λ+ of mutually disjoint sets, and every coloring d:n×n→λ+, there exist a,b∈A with max(a)<min(b) such that c(ai,bj)=d(i,j) for all i,j<n.