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Claim. In Some of my forgotten problems in number theory (Hardy-Ramanujan J. 15 (1992), 34–50; library card erdos_1992_my_forgotten_problems_number_theory), Erdős introduces the problem of Problem 873 as joint with Szemerédi (pp. 47–48) and reports on p. 48 that they proved, for every sequence $A$,

$$F(A,X,3)<c_1X^{1/3}\log X,$$

and that there is a sequence $A$ with $F(A,X,3)>c_2X^{1/3}\log X$ for infinitely many $X$. No proof is given.

Covers. The question for every $\varepsilon>1/3$, with $k=3$ and $X$ large. The second bound shows that $k=3$ gives nothing below the exponent $1/3$.

Standing. Claimed: the paper states the result without proof, so it is not listed as refereed. The site credits the result to Erdős and Szemerédi on a problem it labels OPEN; that credit is commentary, not acceptance.

Depends on. No page of this wiki.