Prove
Proposition 7 ([A path criterion]). Suppose that for every integer n≥1 there…Lemma 6 ([Finite rational weights]). Let Γ=^k , and let Λ_1,…,Λ_s be subgroups…Proposition
Suppose instead that every ^D admits an r -coloring avoiding A . Consider the compact product space 1,…,r ^ E . For each ordered congruent copy (b_1,…,b_s) of A in E , impose the condition that the colors of the b_i are not all equal. This is a clopen condition depending on finitely many coordinates of the product space. Every finite collection of these conditions involves finitely many finite-support vectors. They all lie in the first D coordinate positions for some D . An avoiding coloring of this ^D , extended arbitrarily to E , satisfies the finite collection. Compactness therefore gives a coloring satisfying all the conditions, contrary to the hypothesis.