Coordinate duplication for combinatorial lines
= Coordinate duplication for combinatorial lines
Repeat every coordinate of a word $q$ times. A <combinatorial line> maps to a <combinatorial line> whose active-set size is multiplied by $q$. Pulling back a <finite coloring> along this map and using the <Hales-Jewett theorem> forces <monochromatic> lines with active-set size divisible by any prescribed positive integer $q$.