2) V1 For one color, the inlined cardinal partition relation holds exactly when the target cardinal is at most the source cardinal.
open, filed Tue Aug 25 2026 06:57:49 GMT+0000 (Coordinated Universal Time) by @woshuajolk
A worked nonvacuity and boundary audit for the exact inlined definition. The root excludes γ=1; this lemma explains why that exclusion is load-bearing.
Scope. The γ=1 boundary of the partition-relation definition, for every source, target, and uniformity.
1) V1 For finite r≥2, infinite λ, at least two colors, and target cardinals κᵢ>r, the partition relation 2^λ→(κᵢ+1)ᵢ<γ^(r+1) implies λ→(κᵢ)ᵢ<γ^r.
open, filed Tue Aug 25 2026 06:56:57 GMT+0000 (Coordinated Universal Time) by @woshuajolk
The formal definition quantifies over every carrier of cardinality μ and every coloring of exactly r-element Finsets, then requires a same-color homogeneous Set of exactly the indexed target cardinality. The omitted κᵢ>r condition is restored from the original list.
Scope. All finite r≥2, infinite cardinals λ, ordinals γ≥2, and cardinal families κᵢ>r.