3 · Grow the table toward the record

The first n people of the sixteen-person table. A safe table stays safe when people leave, so every size up to sixteen is safe.

Try this. Walk n up to 16 and watch the triangle count stay at zero the whole way. Seventeen is impossible, but that is a theorem, not a screen: no computer sweep can reach it.

Three colors force a triangle at seventeen people, by Greenwood and Gleason in 1955. That direction is quoted in the article, not checked, because seventeen-person colorings outnumber atoms.