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Source. Published pp. 349–350, Lemma 15; proved through Theorem 16 on pp. 351–354 (published scan).

Statement. Given real $c_1,\ldots,c_k$ and $b\ne0$, some finite coloring of $\mathbb R$ has no monochromatic solution of

$$\sum_{i=1}^k c_i(x_i-x_0)=b.$$

Complete proof. Apply theorem_16 to $F=\mathbb R$ and these coefficients. A monochromatic solution of the displayed equation would be a forbidden paired solution on setting $x_i'=x_0$ for every $i$. This proves the lemma. $\square$

Bears on. #174.