= Canonical arithmetic progression dichotomy
Every colouring of the <positive integers>, even with infinitely many colours, has an <arithmetic progression> of any prescribed finite length on which the colouring is constant or <injective>. Encode a progression by the <equivalence relation> of equality of its colours. This gives a <finite colouring> of its starting point and step. A <Gallai theorem for an integer lattice> copy of a suitable two-dimensional pattern makes either all indices inequivalent, or one equal-colour pair range independently over two whole progressions. In the second case, one of those progressions is <monochromatic>.
Back to article page
