 Solution

Apply the Gallai theorem for an integer lattice to the finite patternAlthough some coordinates are negative, translating $F$ into the positive quadrant and applying the positive-lattice theorem produces a monochromatic copy $F_0=(a_0,d_0)+qF\subseteq {\mathbb{N}}^2$, with integer $q>0$. The translation is absorbed into $(a_0,d_0)$. Let $(r,s)$ be the common position-pair label on this copy.

For each $0\le u,v<m$, the starting point and steplabel the red edge at positions $r,s$. Its endpoints simplify toThus the two arithmetic progressionshave a positive common difference $q(s-r)$, and every cross-pair is red. Their entries are positive because they occur as endpoints of the positive-start, positive-step progressions represented in $F_0$.

It remains to check disjointness, rather than infer it merely from distinct starting points. The pattern contains points whose second coordinate is $-(m-1)$. Since all of $F_0$ lies in ${\mathbb{N}}^2$,ConsequentlyTherefore $A$ and $B$ are disjoint $m$-term arithmetic progressions with every cross-edge red. Combined with the first branch, this proves the dichotomy.