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⊆λ+, such that rts[A{circled asterisk operator}B]⊇C{circled asterisk operator}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.

Original languageEnglish
Pages (from-to)1085-1099
Number of pages15
JournalAdvances in Mathematics
Issue number2
StatePublished - 1 Oct 2012
Externally publishedYes


  • Minimal walks
  • Off-center club guessing
  • Partition relations
  • Square-bracket operation
  • Successor of singular cardinal


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