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Lemma 26 ([Finite prime complexity]). Let x_1,…,x_s be positive real numbers whose pair ratios are positive rationals. Suppose that, for every i<j , x_j/x_i=c_ ij u_ ij ^ ε , c_ ij ∈ C, ε∈1,-1, where C is a fixed finite set of positive …

Lemma 26 ([Finite prime complexity]). Let x1,…,xs be positive real numbers whose pair ratios are positive rationals. Suppose that, for every i<j , xj/xi=cij uijε , cij ∈ C, ε∈1,-1, where C is a fixed finite set of positive rationals, uij is a positive integer with at most h prime factors counted with multiplicity, and ε is the same for all pairs. There is a number of colors depending only on C,h such that on every homogeneous triangle all three x 's are equal. Consequently arbitrarily long such arrays have arbitrarily long constant subarrays.

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Lemma 25 ([Transport of a coordinate stabilizer]). Suppose that the last cover…Lemma 24 ([Twisted tensor maximality]). An almost simple irreducible maximal…观点
Let S contain the primes occurring in the numerators or denominators of members of C . For ℓ∈S , the integer v_ℓ(x_j/x_i) lies in a fixed bounded interval: the contribution of c_ ij is bounded, and that of u_ ij has absolute value at most h . Outside S all valuations have sign ε , and the sum of their absolute values is at most h . Color (i,j) by the valuation vector on S and this last sum. This is a finite coloring with the asserted uniformity. For i<j<k we have v_ℓ(x_k/x_i)=v_ℓ(x_j/x_i)+v_ℓ(x_k/x_j). On a homogeneous triangle each coordinate of the common valuation vector therefore equals twice itself, so it is zero. Outside S there is no cancellation because all valuations have the same sign. Their total absolute value likewise equals twice itself, hence is zero. Every valuation of every pair ratio is zero, so all ratios equal one. Finite Ramsey gives the last assertion.
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