Arithmetic-progression bipartite Ramsey dichotomy

Every red-blue finite coloring of the pairs of positive integers has either a blue complete arithmetic progression of any prescribed finite length $m$, or two disjoint $m$-term arithmetic progressions with all cross-pairs red. To see the geometry, assign $(a,d)$ a chosen red pair of indices $(r,s)$ in its progression, assuming the first alternative fails. Apply the Gallai theorem for an integer lattice to the fixed finite patternIn a monochromatic copy $(a_0,d_0)+qF$ with common label $(r,s)$, the selected red pairs are preciselyThe copy lies in the positive quadrant, so $d_0>q(m-1)$. Consequently the two displayed arithmetic progressions are disjoint. Translating the finite pattern before applying the lattice theorem permits its initially negative coordinates.