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Claim. The answer to Problem 1021 is yes: for every $k\ge3$ there is $c_k>0$ with $\mathrm{ex}(n,G_k)\ll n^{3/2-c_k}$, and one may take $c_k=6^{-k}$. The claimed result is Theorem 5.1 of D. Conlon and J. Lee, On the extremal number of subdivisions, Int. Math. Res. Not. IMRN 2021, no. 12, 9122--9145, DOI 10.1093/imrn/rnz088 (Crossref record, 2026-10-07: published online 3 June 2019, in the issue of 21 June 2021, whose date the paper link carries): for each fixed $t\ge3$ the one-subdivision $H_t$ of $K_t$, in which every edge is replaced by a path of length two with its own internal vertex, satisfies $\mathrm{ex}(n,H_t)\le C_tn^{3/2-6^{-t}}$ for a constant $C_t$ depending on $t$. The corpus states it on the result page Theorem 5.1 of the arXiv:1807.05008v2 manuscript (p. 9; card). The graph $G_k$ of the question is $H_k$: the vertex $z_j$ joined to the pair $\{y_i,y_{i'}\}$ is the internal vertex of the path between $y_i$ and $y_{i'}$, and distinct pairs have distinct $z_j$, as the problem page writes out under Progress. The same paper's Theorem 1.3 gives, without an explicit constant, some $\delta_H>0$ for every fixed $C_4$-free bipartite $H$ with degree at most two on one side, a second route to the question since $G_k$ meets both conditions. The theorem gives no uniform bound when $k$ grows with $n$, and the site's commentary records that Erdős and Simonovits showed, in unpublished work, that $c_k\to0$ as $k\to\infty$ is forced.

Depends on. Nothing in this wiki; the identification $G_k=H_k$ is written on the problem page and carries no independent review.

Acceptance. Refereed publication in International Mathematics Research Notices, cited with its venue above, the refereed evidence. The reviewed evidence is the documented acceptance of the site's curator, Thomas Bloom, who took no part in the paper: the site's label is PROVED and its commentary credits the proof to [CoLe21] with the value $c_k=6^{-k}$; the forum comment of 13 September 2025 pointing to the restatement of the theorem in Conlon, Janzer and Lee's later paper (arXiv:1903.10631, its Theorem 1.3) is marked by the site as addressed, and the curator's comment of 21 January 2026 explains that both resolving papers appeared on arXiv in 2018, this one a few months before Janzer's. The proof-claim tab is empty, and the community database records the problem proved. This page's date is the arXiv v1 posting, 13 July 2018 (arXiv record, 2026-10-07). Read depth: pp. 1--2, 9 and 14 of the manuscript are the basis for the definitions, the statement and the final assembly of the proof; the dependent-random-choice argument of Section 5 and its lemmas are not reconstructed, the journal typesetting is not compared with the manuscript, and nothing is independently reviewed by this project. Janzer's sharper exponent has its own claim page, Janzer.