= 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 pattern
$$
\bigcup_{0\leq r<s<m}\{(si-rj,j-i):0\leq i,j<m\}.
$$
In a <monochromatic> copy $(a_0,d_0)+qF$ with common label $(r,s)$, the selected red pairs are precisely
$$
\bigl(a_0+rd_0+q(s-r)i,\ a_0+sd_0+q(s-r)j\bigr).
$$
The 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.
Back to article page
