 Fixed-active-size obstruction for combinatorial lines

For any proposed positive active size $d$, the displayed two-colour finite colouring of binary words gives opposite colours to the two points of every combinatorial line with exactly $d$ active coordinates. Their counts of the letter $2$ differ by $d$. Thus no dimension can guarantee a monochromatic line of one universally prescribed active size, even though the Hales-Jewett theorem guarantees some active size. The colouring is allowed to depend on $d$ because $d$ is chosen before the colouring.