= Rado solution inside an m-p-c set
{c}
Suppose the columns $a_i$ of a <matrix> over the <rational numbers> satisfy the <columns condition> with blocks $B_1,\ldots,B_s$. Choose rational coefficients $\lambda_{ij}$ such that
$$
\sum_{i\in B_j}a_i+\sum_{i\in B_1\cup\cdots\cup B_{j-1}}\lambda_{ij}a_i=0.
$$
Choose a positive integer $c$ clearing all denominators and $p\geq\max|c\lambda_{ij}|$. For generators of any full positive <M-p-c set>, the coordinates
$$
x_i=cz_h+\sum_{j>h}c\lambda_{ij}z_j\qquad(i\in B_h)
$$
lie in that set and satisfy $Ax=0$. Thus the <monochromatic m-p-c set theorem> supplies the sufficiency direction of <Rado's theorem>.
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