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Claim. The Theorem of Section 3 of Hanani's On quadruple systems (p. 148) states that a system $S(3,4,n)$, a family of $4$-subsets of an $n$-element set containing every $3$-subset exactly once, exists if and only if $n\equiv2$ or $4\pmod 6$. The congruence is the divisibility condition of Problem 722 for $(r,k)=(3,4)$, $\binom{4-i}{3-i}\mid\binom{n-i}{3-i}$ for $i=0,1,2$, so the answer is yes for $(3,4)$ and every $n$, not only large $n$. The necessity is the counting argument of the paper's Section 1; the sufficiency is proved by induction on $n$, through recursive constructions that build a system on $n$ points from systems on smaller admissible numbers of points. The paper settles the quadruple case of Steiner's problem of 1853, open for more than a century; Erdős recounts in [Er81], Part VI, that Hanani told him of the result in 1955 and that he urged its publication.

Covers. The case $(r,k)=(3,4)$ of the problem, for every $n$: the divisibility conditions, $n\equiv2$ or $4\pmod 6$, suffice for a Steiner system $S(3,4,n)$. The paper says nothing about any other pair $(r,k)$; the cases $(2,4)$ and $(2,5)$ are Hanani's block designs, and the general case is Keevash's existence of designs.

Acceptance. Refereed: H. Hanani, On quadruple systems, Canad. J. Math. 12 (1960), 145–157, received 9 February 1959; the record gives the year only, and the page is dated to its first day. The site's commentary credits the case $(3,4)$ to Hanani under its key [Ha61], which the site's reference record resolves to his 1961 paper on $2$-designs; this paper is the one that proves the case, and [Er81] attaches it to the 1961 paper as well. The curator names the case in the progression that ends with Keevash while crediting the problem's PROVED label to Keevash, which does not settle the problem on this result, so the page lists no reviewed evidence. The proof has not been reconstructed or independently reviewed in this corpus.