= Fixed-active-size obstruction for combinatorial lines
{title2=$c(w)=\lfloor\#\{j:w_j=2\}/d\rfloor\bmod2$}
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.
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