Definition 1.6 (Focused). Let $A_{1}, A_{2}, \dots, A_{k}$ be arithmetic progressions of length $m$ . Say

A_{i} = {a_{i}, a_{i} + d_{i}, \dots, a_{i} + (m - 1) d_{i}} .

We say that these arithmetic progressions are focused if $a_{i} + m d_{i} = f$ for all $i \in [k]$ .

If we have a colouring of $ℕ$ and $A_{1}, \dots, A_{k}$ are focused arithmetic progressions if each $A_{i}$ is monochromatic but they all have different colours, then we call them colour-focused.