 Rado solution inside an m-p-c set

Suppose the columns $a_i$ of a matrix over the rational numbers satisfy the columns condition with blocks $B_1,\dots ,B_s$. Choose rational coefficients ${\lambda }_{ij}$ such thatChoose a positive integer $c$ clearing all denominators and $p\ge \max |c{\lambda }_{ij}|$. For generators of any full positive M-p-c set, the coordinateslie in that set and satisfy $Ax=0$. Thus the monochromatic m-p-c set theorem supplies the sufficiency direction of Rado's theorem.