Proof of Rado’s Theorem. Want to prove $A$ is partition regular if and only if it has the column property.

If $A$ is partition regular, then by Proposition 2.5, it has the column property.

For the other direction, let $\tilde{c}$ be a finite colouring of $\mathbb{N}$. Also, since $A$ has column property there exists $m,p,c$ such that $Ax=0$ solutions in any $(m,p,c)$-set. By Theorem 2.6 there exists a monochromatic $(m,p,c)$-set with respect to $\tilde{c}$. But this gives a monochromatic solution to $Ax=0$. □