The existence spectrum of overlarge sets of Kirkman triple systems with two special points (OLKTS$^+$)

Xu and Ji (2024) introduced the structure OLKTS$^+(v)$: an overlarge set of Kirkman triple systems of order $v$ — a partition of all triples of a $(v+1)$-point set into $v+1$ Kirkman triple systems, one on each $v$-point subset — admitting two distinguished special points at which a four-way block-membership equivalence holds. This weakened form of the classical RDSQS$(v+1)$/OLKTS$^\star$ ingredient suffices to construct a large set of Kirkman triple systems LKTS$(3v)$, and its known instances (base orders $9,15,21,33$ plus tripling, product, and fan-based recursive families) cover only a sparse subset of the admissible orders $v \equiv 3 \pmod{6}$. The problem is to determine the complete existence spectrum of OLKTS$^+(v)$ over admissible orders — in particular, whether OLKTS$^+(v)$ exists for every admissible $v$ — with each newly settled order directly yielding an LKTS$(3v)$ through the reduction of Xu and Ji, and thereby advancing the long-open Sylvester existence problem for large sets of Kirkman triple systems.

Background

A Steiner triple system of order $v$, STS$(v)$, is a pair $(X,\mathcal{B})$ where $X$ is a $v$-point set and $\mathcal{B}$ is a set of 3-subsets (blocks) such that every pair of points lies in a unique block. A parallel class is a set of blocks that partitions the point set; an STS$(v)$ whose block set partitions into parallel classes is called resolvable, or a Kirkman triple system KTS$(v)$. A KTS$(v)$ exists if and only if $v\equiv 3\pmod 6$, the solution of Kirkman's schoolgirl problem by Ray-Chaudhuri and Wilson. Since each STS$(v)$ has $v(v-1)/6$ blocks while the point set carries $\binom{v}{3}=v(v-1)(v-2)/6$ triples, at most $v-2$ mutually disjoint STS$(v)$ can sit on the same point set; a large set of Steiner triple systems LSTS$(v)$ — $v-2$ pairwise disjoint STS$(v)$ which then necessarily partition all triples — is such a maximal family, and its spectrum was completely determined by Lu's six-part series and Teirlinck's completion: an LSTS$(v)$ exists for every $v\equiv 1,3\pmod 6$ except $v=7$. A large set of Kirkman triple systems LKTS$(v)$ is an LSTS$(v)$ in which every member is resolvable. This existence problem, posed in the 19th century in connection with Kirkman's schoolgirl problem (as recounted in the surveyed literature below), remains one of the best-known open problems in combinatorial design theory: Wang (2026) records that an LKTS$(v)$ is now known for every admissible order $v<200$ except possibly $v=57$ and $v=87$.

An overlarge set of Kirkman triple systems OLKTS$(v)$ is the one-point-larger analogue: on a $(v+1)$-point set $X$, a collection $\{(X\setminus\{x\},\mathcal{B}_x):x\in X\}$ of $v+1$ KTS$(v)$, one on each $v$-point subset, whose blocks partition all triples of $X$. (By contrast, the spectrum of overlarge sets of ordinary — non-resolvable — Steiner triple systems is completely known, as recorded in Yuan and Kang's study.) Yuan and Kang constructed the first infinite OLKTS families using tripling and product recursions based on the strongest, or star, condition: OLKTS$^\star(v)$ denotes an overlarge set such that for every point $z\in X$ and every triple of not-necessarily-distinct points $a,b,c\in X$ the four membership statements $\{a,b,c\}\in\mathcal{B}_z$, $\{a,b,z\}\in\mathcal{B}_c$, $\{a,z,c\}\in\mathcal{B}_b$, $\{z,b,c\}\in\mathcal{B}_a$ are equivalent. This condition is the overlarge-set counterpart of an RDSQS$(v+1)$, a Steiner quadruple system of order $v+1$ (a $3$-$(v+1,4,1)$ design, i.e. a system of 4-subsets covering every triple exactly once) whose derived triple system at every point is resolvable; RDSQSs have been a standard ingredient in recursive constructions of LKTSs of order $3v$. Even the OLKTS$^\star$/RDSQS spectrum is far from complete.

Xu and Ji (2024) weakened the star condition: an OLKTS$^+(v)$ is an overlarge set of KTS$(v)$ with two distinguished special points $z_1,z_2\in X$ such that the four-way equivalence above holds for each $z_i$, $i\in\{1,2\}$, with no condition imposed at the remaining points. They proved that OLKTS$^+(v)$ fully replaces RDSQS$(v+1)$ in the classical route: from an OLKTS$^+(v)$ one obtains a partitionable resolvable candelabra system (an RPCS — a resolvable design with groups and a stem whose parallel classes cover all triples not meeting the groups) and hence an LKTS$(3v)$. They constructed an explicit OLKTS$^+(33)$ via an algebraic blow-up of a resolvable Steiner $2$-design $S(2,4,16)$ (a design with 4-point blocks covering every pair once) over GF$(16)$, using an RDSQS$(10)$ to supply OLKTS$^+(9)$ fillers, and established fan-design, tripling, and product recursions (the fan-design recursions apply Hartman's fundamental construction, a standard composition technique that inflates a design by replacing its typed blocks with smaller auxiliary designs). The resulting families — OLKTS$^+(2^{2n+1}25^m+1)$ for integers $n,m\ge 1$; OLKTS$^+(6x\pm 3)$, OLKTS$^+(24x+9)$, and OLKTS$^+(12x+3)$, each with $x=x(m,n,u)=5^m(4^u5^n+(4^u-1)/3)$ for nonnegative integers $m,n,u$ (the range $m+n+u\ge 1$ is stated for the $6x\pm 3$ family in Theorem 4.5); the tripling OLKTS$^+(v)\to$ OLKTS$^+(3v)$; and the product OLKTS$^+(u),\text{OLKTS}^+(v)\to$ OLKTS$^+(uv)$ — include the multiplicative family OLKTS$^+(3^{a+b+c}\cdot 5^a 7^b 11^c)$ and yield the LKTS family LKTS$(3^{a+b+c+1}\cdot 5^a 7^b 11^c)$. Since every member of an overlarge set is a KTS$(v)$, the admissible orders are precisely $v\equiv 3\pmod 6$, and the constructed families cover only a sparse subset of these; the source paper establishes only some infinite classes of OLKTS$^+$s and LKTSs, leaving the existence spectrum over the remaining admissible orders open.

Problem Statement

Determine the existence spectrum of special overlarge sets of Kirkman triple systems: for exactly which admissible orders $v\equiv 3\pmod 6$ does there exist an OLKTS$^+(v)$ — a partition of all triples of a $(v+1)$-point set $X$ into $v+1$ Kirkman triple systems $(X\setminus\{x\},\mathcal{B}_x)$, $x\in X$, admitting two distinguished points $z_1,z_2\in X$ such that, for each $i\in\{1,2\}$ and all (not necessarily distinct) $a,b,c\in X$, the four membership statements $\{a,b,c\}\in\mathcal{B}_{z_i}$, $\{a,b,z_i\}\in\mathcal{B}_c$, $\{a,z_i,c\}\in\mathcal{B}_b$, and $\{z_i,b,c\}\in\mathcal{B}_a$ are equivalent? In particular: does an OLKTS$^+(v)$ exist for every admissible order $v$, or are there admissible orders for which no such design exists? Via the reduction of Xu and Ji, each OLKTS$^+(v)$ obtained immediately yields an LKTS$(3v)$, so the spectrum directly controls existence of large sets of Kirkman triple systems at every order $3v\equiv 9\pmod{18}$; this consequence is the motivation for the problem, while the completion boundary is the spectrum determination itself.

References

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