Note A Ramsey Statement for Infinite Groups

Abstract

If κ is a cardinal, n < ω, then there exists an Abelian group G such that if F: G→κ, then there exist distinct elements a_i,α∈G (1≤i≤n,α<κ), and a color τ<κ such that if 1≤i₀<···<i_r≤n, α_r<κ, then $F\left( {{a_{{i_1},{\alpha _1}}} + \cdots + {a_{{i_r},{\alpha _r}}}} \right) = \tau$.