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Claim. There is an explicit set of positive integers with no three-term arithmetic progression whose reciprocal sum exceeds $3.00849$; hence, for the function $f$ of Problem 169,

$$f(3)\ge3.00849.$$

The set takes a translate of the Szekeres set, the greedy set with no three-term progression, up to a finite cutoff and continues it with denser blocks of Behrend's type, as Walker's introduction and Kiichi's manuscript describe it. The site's commentary credits the bound to Wróblewski as the record for $f(3)$; Kiichi's explicit set of 2026, built by gluing further blocks of the same type onto Wróblewski's head, claims the larger value $f(3)\ge3.0085385$ and is pending.

Covers. The lower bound $f(3)\ge3.00849$. Not covered: the value of $f(3)$, any other $k$, and the displayed limit question.

Depends on. No page of this wiki; the result is the paper's.

Acceptance. Refereed: J. Wróblewski, A nonaveraging set of integers with a large sum of reciprocals, Math. Comp. 43 (1984), no. 167, 261–262. The site's curator credits the bound in the problem's commentary, but the site labels the problem OPEN, so that commentary is not acceptance of the problem and the page lists no reviewed evidence.

Dating. The page is dated by the issue of the publisher's record, number 167, July 1984; the day is a placeholder.