## 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{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 language | English |
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Pages (from-to) | 1085-1099 |

Number of pages | 15 |

Journal | Advances in Mathematics |

Volume | 231 |

Issue number | 2 |

DOIs | |

State | Published - 1 Oct 2012 |

Externally published | Yes |

## Keywords

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