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Claim. The number of distinct cycle sets of graphs on $n$ vertices is at most $2^{n-n^{1/2-o(1)}}$. This is the main theorem of R. Nenadov, Improved bound on the number of cycle sets, arXiv:2501.09904, posted 17 January 2025 (the claim's date), revised 22 September 2025, and published in Combinatorial Theory 6 (2026), no. 1, on 2026-04-20; the corpus's card records it as Theorem 1.1 (p. 2) of the v2 preprint, in the form $2^{n-\Omega(\sqrt n/\log^{3/2}n)}$. In the notation of Problem 84, $f(n)\le2^{n-n^{1/2-o(1)}}$, which sharpens Verstraëte's $o(2^{n-n^c})$ with $c\ge0.1$ (claim page). The abstract describes the method as Verstraëte's reduction to counting the cycle sets of Hamiltonian graphs with many chords or large maximum degree, with near-optimal container lemmas as the new ingredient.

Covers. The first assertion, $f(n)=o(2^n)$, already settled by Verstraëte's accepted claim, with a larger saving in the exponent. The second assertion, $f(n)/2^{n/2}\to\infty$, is not addressed.

Depends on. Nothing in this wiki.

Acceptance. The paper is a refereed publication in Combinatorial Theory, volume 6, issue 1 (2026), published online 2026-04-20, which is the refereed evidence. The site's remark names the improvement, but the site labels the problem OPEN, the second assertion being unsettled, so no reviewed evidence is listed. This corpus has not checked the proof.