LAUSR

If 3 ≤ n < ω, V is a vector space over Q with |V | ≤ אn−2, then there is a well ordering of V such that every vector is the last element of only finitely many n-APs. This implies that there is a set mapping f : V → [V ] with no free set which is an n-AP. If, however, |V | ≥ אn−1, then for every set mapping f : V → [V ] <ω there is a free set which is an n-AP. Our starting point is Rado’s theorem which says that each vector space over Q can be colored with countably many colors, so that no arithmetic progression of length 3 (in short, 3-AP) is monocolored (unpublished, but see [1]). There is a general machinery to obtain similar coloring theorems, and in some cases it even gives the existence of a well ordering such that each point is the last element of only finitely many of the configurations in question. In one example, Erdős and Hajnal proved that R can be colored with countably many colors so the no two points in rational distance get the same color, and their argument gives a well ordering < of R such that for each x ∈ R the set {y < x : d(x, y) ∈ Q} is finite (see [2]). The latter implies the former: given a well ordering < as above, the coloring can be constructed by transfinite recursion. The proof of Rado’s theorem goes, however, differently: one fixes a basis {bα : α < κ} of V and if x = n−1