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Claim. For every $r\ge3$, an $r$-uniform hypergraph on $n$ vertices in which no $3r-3$ vertices span three edges has $o(n^2)$ edges. In the authors' notation, $g_n(v,e,r)$ is the largest number of edges of an $r$-uniform hypergraph on $n$ vertices in which the union of any $e$ edges has more than $v$ vertices, and Theorem 1.7 of P. Erdős, P. Frankl and V. Rödl, The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent, Graphs Combin. 2 (1986), no. 1, 113--121, reads: "Suppose $r\geq3$. Then the following hold. $g_n(3r-3,3,r)=o(n^2)$" (display (4), Section 1). Forbidding every $r$-uniform hypergraph with $3r-3$ vertices and $3$ edges therefore leaves $o(n^2)$ edges, so the problem's $d_r(3)$ is at most $3r-3=(r-2)\cdot3+3$. The theorem's second part, display (5), shows that $g_n(3r-3,3,r)/n^c\to\infty$ for every $c<2$, so the bound has no power saving. The proof (Section 4) uses Szemerédi's regularity lemma; the authors note that the case $r=3$ is the theorem of Ruzsa and Szemerédi, recorded on Problem 716. The library card erdos_1986_asymptotic_number_graphs_not_containing_fixed digests the paper.

Covers. The case $e=3$ of Problem 1178 for every $r\ge3$: together with the Brown–Erdős–Sós lower bound $d_r(e)\ge(r-2)e+3$, the theorem gives $d_r(3)=3r-3$, the conjectured value. Nothing for $e\ge4$.

Depends on. The Brown–Erdős–Sós lower bound, which supplies the matching lower half $d_r(3)\ge3r-3$.

Acceptance. Refereed publication in Graphs and Combinatorics (the publisher's record: volume 2, issue 1, pp. 113--121, issued December 1986 with no day recorded; this page is dated the issue's first day). The site labels the problem OPEN, so its commentary crediting the theorem with $d_r(3)=3r-3$ is not listed as reviewed evidence. The text cited is the scan in the Rényi Institute's Erdős archive, https://users.renyi.hu/~p_erdos/1986-17.pdf. This claim is partial, so the problem stays open.