Existence of large sets of Kirkman triple systems of orders 57 and 87: the smallest open cases of the LKTS existence problem

A large set of Kirkman triple systems, LKTS( $v$ ), is a partition of all triples of a $v$ -element set into $v-2$ mutually disjoint Steiner triple systems, each of which is resolvable into parallel classes. Necessarily $v \equiv 3 \pmod{6}$ , and the existence of an LKTS( $v$ ) for every such $v$ is a long-standing open problem of combinatorial design theory going back to Sylvester's schoolgirl problem. Wang (2026) records that an LKTS( $v$ ) is known for every admissible $v < 200$ except possibly $v = 57$ and $v = 87$ , which are therefore the smallest unresolved instances. The derived problem is to determine whether an LKTS(57) and an LKTS(87) exist.

Background

A Steiner triple system of order $v$ , STS( $v$ ), is a pair ( $X$ , $\mathcal{B}$ ) where $X$ is a $v$ -element set and $\mathcal{B}$ is a collection of 3-element subsets of $X$ , called blocks, such that every 2-element subset of $X$ lies in exactly one block. An STS( $v$ ) is called resolvable if its blocks can be partitioned into parallel classes, each parallel class being itself a partition of the point set $X$ ; a resolvable STS( $v$ ) is called a Kirkman triple system, KTS( $v$ ). It is classical that a KTS( $v$ ) exists if and only if $v \equiv 3 \pmod{6}$ . Two STS( $v$ ) on the same point set are disjoint if they share no block. Since the complete set of triples has $\binom{v}{3}$ members and each STS( $v$ ) contains $v(v-1)/6$ of them, at most $v-2$ mutually disjoint STS( $v$ ) can coexist on a fixed $v$ -set; a partition of all triples into $v-2$ disjoint STS( $v$ ) is called a large set of Steiner triple systems, LSTS( $v$ ). The spectrum of LSTSs is completely determined: an LSTS( $v$ ) exists if and only if $v \equiv 1, 3 \pmod{6}$ and $v \neq 7$ , with the determination completed by Teirlinck (1991). A large set of Kirkman triple systems, LKTS( $v$ ), is an LSTS( $v$ ) in which every constituent STS( $v$ ) is resolvable; admissibility again forces $v \equiv 3 \pmod{6}$ . Whether an LKTS( $v$ ) exists for every admissible $v$ — a question raised in connection with Kirkman's and Sylvester's nineteenth-century schoolgirl problem, with early cases settled by Kirkman (order 9, 1850) and Denniston (order 15, 1974) — is one of the best-known open problems in combinatorial design theory.

Known LKTSs come from two kinds of constructions, surveyed by Chang and Zhou (2013). Direct constructions, mostly finite-field or difference-set based, cover a sparse set of orders (for example 9, 15, 21, 33, 39, 51, 69, together with further sporadic and family orders). Recursive constructions include Denniston-type tripling and product constructions, and a program based on Steiner quadruple systems with resolvable derived designs (an RDSQS( $v+1$ ) is a Steiner quadruple system, that is, a 4-uniform Steiner system SQS( $v+1$ ), whose derived design at every point is a resolvable STS( $v$ )); the existence of an RDSQS( $v+1$ ) implies an LKTS( $3v$ ). This program has produced infinite families: LKTS( $2^{2n+1}+1$ ) by Xu and Ji (2021), further RDSQS-based families by Liu and Lei (2023) and Tan and Zhou (2025), and, most recently, the construction of Wang (2026), which builds an LKTS( $7q+2$ ) from an LKTS( $q+2$ ) together with an auxiliary permutation structure on an abelian group of order $q$ (a cubic orthomorphism, shown to exist whenever $q$ is a product of prime powers each congruent to 1 modulo 6), yielding orders 93, 177, 219, 261, and 303. Xu and Ji (2024) additionally develop overlarge sets of Kirkman triple systems with an extra property, OLKTS $^+(v)$ , as an intermediate object playing the role of RDSQSs in constructing LKTS( $3v$ ).

As recorded by Wang (2026), combining all known constructions gives an LKTS( $v$ ) for every admissible order $v < 200$ except possibly $v = 57$ and $v = 87$ , which are thus the two smallest unresolved instances of the existence problem. These two orders resist the standard machinery: writing $v = q + 2$ gives $q = 55 = 5 \cdot 11$ and $q = 85 = 5 \cdot 17$ , neither of which is a prime power or a product of prime powers each congruent to 1 modulo 6, so the finite-field-based direct constructions do not apply; and $57 = 3 \cdot 19$ , $87 = 3 \cdot 29$ where the cofactors 19 and 29 are not admissible KTS orders (they are not congruent to 3 modulo 6), so tripling-type constructions LKTS( $v$ ) $\to$ LKTS( $3v$ ) cannot reach them either. The source paper therefore proposes reaching 57 and 87 by direct construction (possibly with a prescribed automorphism group) or by new recursions of the forms LKTS( $v$ ) $\to$ LKTS( $4v-3$ ) (order 57 from the known LKTS(15)) and LKTS( $v$ ) $\to$ LKTS( $4v+3$ ) (order 87 from the known LKTS(21)), possibly via intermediate objects weaker than an LKTS such as overlarge sets, Kirkman frames (resolvable group divisible structures with holes), systems with holes, or Steiner quadruple systems with resolvable derived designs.

Problem Statement

Determine whether there exists a large set of Kirkman triple systems of order 57 — that is, 55 mutually disjoint Steiner triple systems on a common 57-point set, each resolvable into 28 parallel classes, together partitioning all $\binom{57}{3} = 29260$ triples — and whether there exists an LKTS(87) — 85 mutually disjoint resolvable Steiner triple systems on an 87-point set, each resolved into 43 parallel classes, partitioning all $\binom{87}{3} = 105995$ triples.

These are the two smallest admissible orders ( $v \equiv 3 \pmod 6$ ) for which the LKTS existence problem is unresolved, and every admissible order below 200 other than these two is settled. The broader question — whether an LKTS( $v$ ) exists for every $v \equiv 3 \pmod 6$ — remains open and is the context of this problem.

References

Chen Wang (2026), A new recursive construction for large sets of Kirkman triple systems, arXiv:2607.21521, https://arxiv.org/abs/2607.21521
Yan Liu, Jianguo Lei (2023), More results on large sets of Kirkman triple systems, Designs, Codes and Cryptography 91(8), 2677-2686, doi:10.1007/s10623-023-01221-9, https://doi.org/10.1007/s10623-023-01221-9 (preprint arXiv:2210.06214, https://arxiv.org/abs/2210.06214)
Yuli Tan, Junling Zhou (2025), Steiner quadruple systems with minimum colorable derived designs: constructions and applications, Designs, Codes and Cryptography 93(10), 4117-4140, doi:10.1007/s10623-025-01677-x, https://doi.org/10.1007/s10623-025-01677-x (also arXiv:2503.11934, https://arxiv.org/abs/2503.11934)
Juanjuan Xu, Lijun Ji (2024), Special overlarge sets of Kirkman triple systems, Designs, Codes and Cryptography 92(8), 2267-2285, doi:10.1007/s10623-024-01386-x, https://doi.org/10.1007/s10623-024-01386-x
Juanjuan Xu, Lijun Ji (2021), Large sets of Kirkman triple systems of orders 2^{2n+1}+1, Discrete Mathematics 344(6), 112373, doi:10.1016/j.disc.2021.112373, https://doi.org/10.1016/j.disc.2021.112373
R. H. F. Denniston (1974), Sylvester's problem of the 15 schoolgirls, Discrete Mathematics 9(3), 229-233, doi:10.1016/0012-365X(74)90004-1, https://doi.org/10.1016/0012-365X(74)90004-1
Yanxun Chang, Junling Zhou (2013), Large sets of Kirkman triple systems and related designs, Journal of Combinatorial Theory, Series A 120(3), 649-670, doi:10.1016/j.jcta.2012.12.001, https://doi.org/10.1016/j.jcta.2012.12.001
Luc Teirlinck (1991), A completion of Lu's determination of the spectrum for large sets of disjoint Steiner triple systems, Journal of Combinatorial Theory, Series A 57(2), 302-305, doi:10.1016/0097-3165(91)90053-J, https://doi.org/10.1016/0097-3165(91)90053-J
Svetlana Topalova, Stela Zhelezova (2025), The cyclically resolvable Steiner triple systems of order 57, Mathematics 13(2), 212, doi:10.3390/math13020212, https://doi.org/10.3390/math13020212