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Claim. Let $d_1<d_2<\cdots$ be the $3$-smooth numbers, the integers $2^a3^b$, and let $f(r)$ be the size of the largest subset of $\{d_1,\ldots,d_r\}$ containing no triple $\{n,2n,3n\}$. Then the limit of Problem 168 exists and

$$\lim_{N\to\infty}\frac{F(N)}{N} =\frac13\sum_{r\ge1}f(r)\Bigl(\frac1{d_r}-\frac1{d_{r+1}}\Bigr) =\frac13\sum_{k\in K}\frac1{d_k}, \qquad K=\{k: f(k)>f(k-1)\}.$$

This is Section 4, equations (11) and (12), pp. 106–108, of the paper of Graham, Witsenhausen and Spencer: a set is free of such triples exactly when its intersection with each class $\{t\cdot 2^a3^b\}$, $(t,6)=1$, is, which reduces the extremal count to the function $f$ on the $3$-smooth numbers. The paper tabulates $f(k)$ for $k\le36$ and the first terms of $K$.

Covers. The existence of the limit and its series form. Not covered: a closed form for the value, since the authors see no simple way to determine $K$; the site's commentary reports that Eberhard evaluated the limit from this formula as $0.800965\cdots$, a computation that has no page. Also not covered: the question whether the limit is irrational, which the paper itself raises and the problem repeats.

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

Standing. Claimed. The paper appeared in the collected volume Number Theory and Algebra (Academic Press, New York, 1977), pp. 103–109, which is not shown to be refereed, so the page lists no refereed evidence. The site labels the problem OPEN and its commentary credits the result, as does the formal-conjectures statement file (erdos_168.variants.limit_exists, tagged solved with this attribution and left as sorry); neither is acceptance of the problem.

Dating. The page is dated by the publication year; the volume gives no day, and the day in the page name is a placeholder. The page name follows the offprint's author order; the site lists the authors as Graham, Spencer and Witsenhausen.