## Abstract

Erdös, Hajnal and Rado asked whether N_{2} N_{1} → N_{0} N_{0} N_{1} N_{1}. We show that N_{2} N_{1} {not right arrow} N_{0} N_{0} N_{1} N_{1} is consistent with ZF + GCH. The above symbol means that there exists a coloring of pairs (αβ), α ∈ ω_{2}, β ∈ ω_{1}, by two colors such that each A × B, where A ⊆ ω_{2}, B ⊆ ω_{1}, |A| = א_{0}. |B| = א_{1}, has a pair of each color. N_{2} N_{1} {not right arrow} N_{0} N_{0} N_{1} N_{1} is implied by the following statement (*): There are sets A_{αη} ⊆ ω_{1}, α ∈ ω_{2}, η ∈ ω_{1}, such that (∀α, η, ξ) (η ≠ ξ → A_{αη} ∩ A_{αξ} - 0), (∀α) ∪ {A_{αη}: η ∈ ω_{1}} = ω_{1}, and for every sequence of distinct α_{n} ∈ ω_{2} and every sequence of (not necessarily distinct) η_{n} ∈ ω_{1}, |ω_{1} - ∪ {Aα_{n}η_{n} : η ∈ ω} | ≤ א_{0}. Another consequence of (*) is. For every א_{1} nonprincipal countably additive ideals l_{ρ} ⊆ P(ω_{1}), ρ ∈ ω_{1}, there is a set X ⊆ ω_{1} such that for all ρ ∈ ω_{1} neither X nor ω_{1} - X ∈ l_{ρ}. This shows the independence of a problem of Ulam.

Original language | English (US) |
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Pages (from-to) | 51-59 |

Number of pages | 9 |

Journal | Discrete Mathematics |

Volume | 2 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1972 |