On a problem of Erdös, Hajnal and Rado

Erdös, Hajnal and Rado asked whether N₂ N₁ → N₀ N₀ N₁ N₁. We show that N₂ N₁ {not right arrow} N₀ N₀ N₁ N₁ is consistent with ZF + GCH. The above symbol means that there exists a coloring of pairs (αβ), α ∈ ω₂, β ∈ ω₁, by two colors such that each A × B, where A ⊆ ω₂, B ⊆ ω₁, |A| = א₀. |B| = א₁, has a pair of each color. N₂ N₁ {not right arrow} N₀ N₀ N₁ N₁ is implied by the following statement (*): There are sets A_αη ⊆ ω₁, α ∈ ω₂, η ∈ ω₁, such that (∀α, η, ξ) (η ≠ ξ → A_αη ∩ A_αξ - 0), (∀α) ∪ {A_αη: η ∈ ω₁} = ω₁, and for every sequence of distinct α_n ∈ ω₂ and every sequence of (not necessarily distinct) η_n ∈ ω₁, |ω₁ - ∪ {Aα_nη_n : η ∈ ω} | ≤ א₀. Another consequence of (*) is. For every א₁ nonprincipal countably additive ideals l_ρ ⊆ P(ω₁), ρ ∈ ω₁, there is a set X ⊆ ω₁ such that for all ρ ∈ ω₁ neither X nor ω₁ - X ∈ l_ρ. This shows the independence of a problem of Ulam.