Claims

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1992_01_01_erdos_szemeredi: Erdős reports in 1992, without proof, that he and Szemerédi proved F(A,X,3) < c X^{1/3} log X for every sequence and found a sequence meeting this bound for infinitely many X.

2026_04_29_nayak: Ritvik Nayak's 2026 note, checked with GPT 5.4 Thinking, proves F(A,X,11) << X^{223/676} log X and F(A,X,6) << X^{437/1320} log X, so the question holds for every epsilon above 223/676.

2026_04_30_old_bielefelder: Two 2026 notes posted by old-bielefelder, written by ChatGPT 5.5 Thinking and ChatGPT 5.6 Sol, prove that for every epsilon above 1/4 some k gives F(A,X,k) << X^epsilon uniformly in A; the range up to 1/4 is not covered.

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