µ, θ, χ ≤ κ are cardinals. U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ]2 → θ such that, for every χ < χ, for every A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets, and for every i < θ, there is B ⊆ A of size µ such that, for all distinct a, b ∈ B, we have min(c[a × b]) ≥ i. • U(κ, 2, θ, 2) is a strong negation of κ → (κ)2 θ . • If µ∗ ≥ µ and χ∗ ≥ χ, then U(κ, µ∗, θ, χ∗) implies U(κ, µ, θ, χ). There is no such monotonicity in the third coordinate. • Instances of this principle are implicit in previous work of, for instance, Galvin, Todorcevic, and Shelah.