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Claim. Part III of Wilson's An existence theory for pairwise balanced designs, subtitled Proof of the existence conjectures, states in its abstract that for positive integers $k$ and $\lambda$ a balanced incomplete block design on $v$ points with block size $k$ and index $\lambda$ exists for all sufficiently large $v$ satisfying $\lambda(v-1)\equiv0\pmod{k-1}$ and $\lambda v(v-1)\equiv0\pmod{k(k-1)}$, and that the same holds for pairwise balanced designs whose block sizes lie in a given set $K$. For $\lambda=1$ the congruences are the divisibility conditions of Problem 722 for $r=2$, $(k-1)\mid(n-1)$ and $k(k-1)\mid n(n-1)$, so the answer is yes for $(r,k)=(2,k)$ and every $k$: there is $n_0(k)$ such that every $n>n_0$ satisfying them carries a Steiner system $S(2,k,n)$. Parts I and II (1972) build the theory the proof uses, composition theorems for pairwise balanced designs and the structure of the sets of orders closed under them, and Part II states the existence conjectures that Part III proves. Erdős states the theorem in [Er81], Part VI, with the block count $\binom n2\binom k2^{-1}$, and asks there whether it extends to every $r$, the question the problem poses. The site credits the case $(2,k)$ to Wilson under its key [Wi72], Part II.

Covers. The case $r=2$ of the problem for every $k$: for fixed $k$ and all large $n$ the divisibility conditions suffice for a Steiner system $S(2,k,n)$. The theorem says nothing about $r\ge3$; the general case is Keevash's existence of designs.

Acceptance. Refereed: R. M. Wilson, An existence theory for pairwise balanced designs. III. Proof of the existence conjectures, J. Combin. Theory Ser. A 18 (1975), no. 1, 71–79, completing Parts I and II, J. Combin. Theory Ser. A 13 (1972), 220–245 and 246–273; the issue of Part III is dated January 1975, and the page is dated to the first day of that month. The site's curator names Wilson's 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.