= Run-count obstruction to interval-active lines
For any $N\geq1$, color $w\in[3]^N$ by its first letter together with its number of <runs of a word> modulo $3$. This <finite coloring> uses at most nine colors and has no <monochromatic> <combinatorial line> whose active coordinates form an interval. An active interval starting at the first coordinate changes the first letter. Otherwise, its left boundary and optional right boundary contribute $0$ versus $1$, $0$ versus $2$, or $1$ versus $2$ to the run count as the active letter varies. These differences remain nonzero modulo $3$. The first-letter component is essential for intervals that occupy the entire word.
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