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Claim. Write $\tau_\perp(n)$ for the number of indices $i$ with $(d_i,d_{i+1})=1$ among the divisors $1=d_1<\cdots<d_{\tau(n)}=n$, the function of Problem 1100 (the paper calls it $f(n)$). P. Erdős and G. Tenenbaum, Sur les fonctions arithmétiques liées aux diviseurs consécutifs, J. Number Theory 31 (1989), no. 3, 285--311, prove two bounds that settle the problem's first two questions. Its Corollaire 2 gives the normal order

$$(\log n)^{\log3-1+o(1)}<\tau_\perp(n)<(\log n)^{\log2-\frac12+o(1)}$$

for almost all $n$, the lower bound from Théorème 3 of the paper and the upper bound from the authors' earlier work; since $\omega(n)\sim\log\log n$ for almost all $n$, the ratio $\tau_\perp(n)/\omega(n)$ tends to infinity on a set of density one, which answers the first question in the affirmative. Its Théorème 4 gives the maximal order

$$\max_{n\le x}\tau_\perp(n) \ge\exp\Bigl(\bigl((\log2)^2+o(1)\bigr)\frac{\log x}{(\log\log x)^2}\Bigr) \qquad(x\ge3),$$

a bound Erdős had stated in 1985 [Er85] with a sketch; since $\log x/(\log\log x)^2$ is not $(\log x)^{o(1)}$, there are infinitely many $n$ with $\tau_\perp(n)\ge\exp((\log n)^{1-o(1)})$, and the second question, whether $\tau_\perp(n)<\exp((\log n)^{o(1)})$ for all $n$, has the answer no. The paper also makes the Erdős--Simonovits upper bound for the squarefree extremal function explicit: its Théorème 1, which it says develops an unpublished argument of Erdős and Simonovits, gives $\tau_\perp(n)\ll\tau(n)^{1-c}$ for squarefree $n$ with $c=\tfrac53-\log3/\log2=0.0817\ldots$, that is $g(k)\le(3\cdot2^{-2/3}+o(1))^k=(1.8898\ldots)^k$, while its Théorème 2 shows $\tau_\perp(n)/\tau(n)\ge1/(2\Omega(n))$ for infinitely many $n$.

Covers. The first question (yes: $\tau_\perp(n)/\omega(n)\to\infty$ for almost all $n$) and the second question (no: the bound $\exp((\log n)^{o(1)})$ fails for infinitely many $n$). The claim value is answered because the claim answers the first question yes and the second no. The third question, the growth of $g(k)$, is not determined: the paper's explicit constant bounds $g(k)^{1/k}$ above by $3\cdot2^{-2/3}$ and leaves the gap to the lower bounds, for which see the consequence $g(k)\ge F_{k+1}$ of Chevyrev, Searles and Slinko's theorem drawn on the problem page and Ross's pending golden-ratio bound. The maximal-order question Erdős asked in [Er85], display (27), whether $\tau_\perp(n)<\exp(\epsilon\log n/\log\log n)$ for every $\epsilon>0$ and large $n$, is not in the site's statement and is not covered; it is the subject of Korsky's pending claim.

Acceptance. Refereed: the paper appeared in the Journal of Number Theory, volume 31, issue 3 (March 1989), communicated by R. L. Graham and received 10 December 1987; the Crossref record of its DOI gives these data. Read depth: the introduction's numbered theorems, from the author-archive scan (second link); the proofs are not checked. The site's problem page (last edited 19 October 2025) does not cite the paper and lists the problem as open.