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For $m=1$ the conclusion is immediate, so suppose $m\ge 2$ and index the terms of an arithmetic progression by $0,\dots ,m-1$. There are only finitely many equivalence relations on this index set. Consequently, recording which terms have equal colours produces a finite colouring of the positive parameter pairs $(a,d)$, even though the original colouring need not be finite.

We use the Gallai theorem for an integer lattice in this form: a finite colouring of the positive integer lattice contains a monochromatic homothetic copy of any prescribed finite integer pattern. Patterns with negative coordinates are allowed here by first translating them into the positive quadrant; the translation is absorbed into the base point of the resulting copy.

Consider the finite patternThe Gallai theorem for an integer lattice supplies integers $a_0,d_0$ and $q>0$ such that every point of $(a_0,d_0)+qF$ lies in the positive quadrant and has the same equality equivalence relation $E$. If $E$ has only singleton classes, any parameter pair in this copy gives an arithmetic progression whose colours are pairwise distinct, so the original colouring is injective on it.

Otherwise choose $r<s$ with $rEs$. For every $u,v\in \{0,\dots ,m-1\}$, the parameter pairhas relation $E$. Its terms at indices $r$ and $s$ are therefore equally coloured. But these terms simplify toThus $c(A_u)=c(B_v)$ for every $u,v$. Fixing $v=0$ proves that all $A_u$ have the same colour. Every $A_u$ is positive, because it is a term of one of the positive-parameter progressions above, and the common difference $q(s-r)$ is positive. Hence $(A_u{)}_{u=0}^{m-1}$ is a monochromatic arithmetic progression of length $m$.

The canonical arithmetic progression dichotomy is therefore constant or injective on an arithmetic progression of the prescribed length. The key point is that one equal-colour index pair has been made to vary independently over two complete progressions; a merely monochromatic block of parameter pairs would not by itself give this conclusion.