Claims

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1847_01_01_kirkman: Kirkman (1847) constructs, for every n congruent to 1 or 3 mod 6, a set of triples on n symbols containing every pair exactly once, the case (r, k) = (2, 3) of the problem; a partial answer, refereed.

1960_01_01_hanani: Hanani (1960) proves that a Steiner quadruple system S(3,4,n) exists if and only if n is congruent to 2 or 4 mod 6, the case (r, k) = (3, 4) of the problem; a partial answer, refereed.

1961_06_01_hanani: Hanani (1961) proves that the divisibility conditions suffice for a 2-design with blocks of size 3 or 4 and any index, and of size 5 for index 1, so for the cases (r, k) = (2, 4) and (2, 5); partial, refereed.

1975_01_01_wilson: Wilson (1972-1975) proves that for every k the divisibility conditions suffice for a Steiner system S(2,k,n) once n is large, the case r = 2 of the problem for every k; a partial answer, refereed.

2014_01_15_keevash: Keevash's 2014 theorem that for fixed k > r the divisibility conditions suffice for a Steiner system with parameters (n, k, r) once n is large, settling the existence conjecture; accepted on the site curator's credit.

2016_11_21_glock_kuhn_lo_osthus: An independent proof, by iterative absorption, that the divisibility conditions suffice for an F-decomposition of a large complete r-uniform hypergraph for every F, hence for Steiner systems; refereed in 2023.

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