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Claim. The answer to Problem 926 is yes: for every fixed $k\ge4$, $\mathrm{ex}(n;H_k)\ll_k n^{3/2}$, and the implied constant can be taken linear in $k$. The claimed result is Theorem 6.1 of N. Alon, M. Krivelevich and B. Sudakov, Turán numbers of bipartite graphs and related Ramsey-type questions, Combin. Probab. Comput. 12 (2003), no. 5--6, 477--494 (Section 6, "Improved bounds on a Turán-type problem", pp. 491--493, Theorem 6.1 on p. 491): for integers $k,t\ge2$ and $s\ge1$, the bipartite graph $L_t^{k,s}$ with vertices $x_0$, $y_1,\dots,y_k$ and $x_I^\alpha$ ($I$ a $t$-element subset of $\{1,\dots,k\}$, $1\le\alpha\le s$), in which $y_i$ is joined to $x_0$ and to every $x_I^\alpha$ with $i\in I$, satisfies

$$\mathrm{ex}(2n;L_t^{k,s})\le2^{1+1/t}(s+1)^{1/t}kn^{2-1/t}.$$

The paper notes that for $s=1$ and $t=2$ this graph is the induced subgraph on the first three layers of the Boolean $k$-cube, which is the problem's $H_k$ of the problem page's precise Statement ($x_0$ is $x$, and $x_{\{i,j\}}^1$ is the pair vertex $z_{\{i,j\}}$). At $t=2$, $s=1$ the bound reads $\mathrm{ex}(2n;H_k)\le4kn^{3/2}$; since the extremal number is nondecreasing in the number of vertices, every $N$ has $\mathrm{ex}(N;H_k)\le4k\lceil N/2\rceil^{3/2}$ (authored, one line). For fixed $k$ this is the $O_k(n^{3/2})$ asked for, and it is the bound $\mathrm{ex}(n;H_k)\ll kn^{3/2}$ that the site's commentary credits to the paper. The paper presents the theorem as an improvement of Füredi's $\mathrm{ex}(n;L_t^{k,s})=O((s+1)^{1/t}k^{2-1/t}n^{2-1/t})$, whose case $t=2$ is the first proof of the answer yes (Füredi's claim page), and says that the dependence on $k$ is essentially optimal for $k=n^{1/t}$. The proof splits the vertex set into two halves keeping at least half the edges and, on the bipartite subgraph between them, runs its own random common-neighborhood argument with a weighted count of $t$-subsets (a random $t$-tuple in one half, its common neighborhood in the other, a random $k$-subset of that, then a greedy choice of the vertices $x_I^\alpha$), without invoking Lemma 2.1; it is independent of Füredi's set-system argument.

Acceptance. Refereed publication in Combinatorics, Probability and Computing (Crossref record accessed: volume 12, issue 5--6, pp. 477--494, issued November 2003, whose nominal first day is this page's date; published online 3 December 2003). The site's curator, Thomas Bloom, labels the problem proved and credits the paper with the improvement to $\mathrm{ex}(n;H_k)\ll kn^{3/2}$, but the site's entry carries additional thanks to Noga Alon, so the curator's credit is not listed as reviewed evidence independent of the claimants. The source has a library source card, with the result page Theorem 6.1. Read depth: Theorem 6.1 and the definition of $L_t^{k,s}$; the proof read for structure only; nothing is independently reviewed by this project. The formal-conjectures statement file for the problem states this bound as the variant erdos_926.variants.aks, with no proof link; it is a statement, not a formalization, and is described on the problem page. The acceptance rests on the refereed publication.