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Claim. Let $g(k)$ be the extremal function of Problem 1100, the largest number of coprime consecutive divisor pairs of a squarefree $n$ with $\omega(n)=k$. Michael M. Ross's manuscript An Unconditional Golden-Ratio Lower Bound for Coprime Adjacent Divisors of Squarefree Integers (dated 5 August 2026, Zenodo record of 4 August 2026, DOI 10.5281/zenodo.21790847, CC BY 4.0) proves (its Theorem 1.1) that for an absolute constant $c>0$ and all large $k$

$$g(k)\ge c\,\frac{\varphi^k}{\sqrt k},\qquad \varphi=\frac{1+\sqrt5}{2},$$

so $\liminf_{k\to\infty}g(k)^{1/k}\ge\varphi=1.618\ldots$, above the $\sqrt2$ of the Erdős--Simonovits bound reported in [Er81h]. The divisors of a squarefree $n=p_1\cdots p_k$ are indexed by subsets of $\{1,\ldots,k\}$ ordered by $\sum_{i\in S}\log p_i$, and two divisors are coprime exactly when their subsets are disjoint. Taking the primes from a short block $[Q,(1+\eta)Q]$ with $Q>(1+\eta)^k$ separates the subsets by cardinality, and two limits, $Q\to\infty$ then $\eta\to0$, using only the prime number theorem, transfer the count to a model of subset sums of independent uniform weights within one cardinality layer $j=\lfloor\beta k\rfloor$. There a first-moment estimate shows that the expected number of consecutive disjoint pairs is $\gg_\beta\binom{k-j}{j}$: blockers using a coordinate outside the pair are handled by smoothing and a hypergeometric tail bound, and blockers inside the pair's union are handled by splitting the gap into two independent signed sums, so that blocking costs the square of the window length. The single layer $\beta=(5-\sqrt5)/10$ gives $\binom{k-j}{j}\asymp\varphi^k/\sqrt k$.

Submission note. Posted to erdosproblems.com as a proof claim by Michael M. Ross (account michaelmross) on 5 August 2026, giving "Claude Fable 5 (Anthropic) and GPT-5.6 Sol (OpenAI)" as the AI used:

An improvement to the Erdős–Simonovits lower bound of (√2 + o(1))^k. The main idea is to reframe how the divisors of a squarefree integer are ordered, turning it into a problem about disjoint subset sums with random weights. Looking at a fixed cardinality layer, two disjoint sums should ideally be next to each other with a probability of about 1/binomial(k,j), though other sums—which act as blockers—can end up in between them. While standard density smoothing handles external blockers, a fresh idea deals with internal ones, using a swap decomposition to break the gap down. Because blocking forces two small-gap conditions to happen at the same time, this gives us a crucial theta-squared factor. Picking that one optimal layer j at roughly 0.276393k gives us the exact exponential base φ≈1.618034. Notes: Notes on the proof: https://michaelmross.github.io/notes_on_the_proof.html Visual companion: https://michaelmross.github.io/within_layer_companion.html

Covers. A lower bound for the third question only: the growth of $g(k)$ is at least $\varphi^{k-o(k)}$. It does not determine $g(k)$, whose upper bound stands at $(3\cdot2^{-2/3}+o(1))^k$ by the accepted Erdős--Tenenbaum page, and it does not bear on the first two questions.

Authorship and systems. Michael M. Ross, an independent researcher posting under the forum account michaelmross, submitted the claim on 2026-08-05; the forum's tab names Claude Fable 5 (Anthropic) and GPT-5.6 Sol (OpenAI), and the manuscript's acknowledgments thank the same two systems for their contributions. The manuscript consolidates the author's earlier Zenodo notes, which it cites, including a conditional version of the bound and a separate within-layer lemma; the author's companion pages, notes on the proof and a visual companion, are linked from the forum entry.

Standing. Posted on the problem's forum as a partial proof claim on 2026-08-05 with the Zenodo record as its external link; the claim carried no comments. On the discussion thread, a comment of 2026-07-30 by Samuel Korsky reported that a GPT-5.6-assisted literature review found a gap in a uniform estimate of an earlier Ross preprint, so that its golden-ratio bound should then be called conditional, and a comment of the same day, also by Korsky, said that a $\varphi^k$ lower bound follows from a 2011 arXiv paper on flippable pairs. That remark is correct: Theorem 3 of Chevyrev, Searles and Slinko (Order 30 (2013), 749--761; arXiv:1103.3938) gives a comparative probability order on $k$ atoms, representable by integer weights, with $F_{k+1}$ flippable pairs, each a pair of disjoint sets adjacent in the order (their Definitions 4 and 5); primes whose logarithms approximate a large multiple of the weights turn these into coprime consecutive divisors of a squarefree $n$ with $\omega(n)=k$, so $g(k)\ge F_{k+1}\gg\varphi^k$, stronger than this manuscript's bound by a factor $\sqrt k$. The manuscript, which postdates both remarks, does not cite that paper; the remark on the earlier preprint's gap has not been checked against its source. The site's curator has not commented, the problem's label is unchanged, the manuscript is not refereed and nobody has recorded accepting it, so the claim is claimed.