Completing the existence spectrum of overlarge sets of Kirkman triple systems: the residual OLKF(6^k) and 3-OLGKS(6^{k-1},4) checklist of Yuan and Kang

An overlarge set of Kirkman triple systems, OLKTS(v), is a partition of all triples of a (v+1)-element set into v+1 Kirkman triple systems of order v, each living on the point set with one point deleted; it can exist only when $v \equiv 3 \pmod 6$. Yuan and Kang (2009) proved a conditional reduction (their Theorem C): if there exist both an overlarge set of Kirkman frames OLKF(6^k) and a 3-OLGKS(6^{k-1},4) for every $k \in \{6,7,\dots,40\}\setminus\{8,17,21,22,25,26\}$, then an OLKTS(v) exists for every $v \equiv 3 \pmod 6$ with $v \neq 21$. Subsequent literature shows: (i) the exceptional order $v=21$ has since been settled affirmatively (an explicit RDSQS(22) of Liu and Lei, 2023, yields OLKTS(21) by taking derived designs); (ii) the finite checklist itself remains entirely open — no later work constructs OLKF(6^k) or 3-OLGKS(6^{k-1},4) for any of the residual k, and the OLKTS spectrum is still incomplete as of 2026 (for instance OLKTS(51), OLKTS(69), OLKTS(75) are unknown). The problem is therefore to settle the residual checklist — construct the required Kirkman-frame overlarge sets and OLGKS modules, or prove some cannot exist — thereby completing, together with the now-settled order 21, the existence classification of OLKTS(v) for all admissible v.

Background

A Steiner triple system STS(v) is a pair $(X,\mathscr{B})$ where $X$ has v points and $\mathscr{B}$ is a collection of 3-subsets (blocks) such that every 2-subset of $X$ lies in exactly one block; it exists if and only if $v \equiv 1,3 \pmod 6$. A Kirkman triple system KTS(v) is a resolvable STS(v): its blocks partition into parallel classes, each of which partitions the point set. Resolvability forces $v \equiv 3 \pmod 6$. A large set LKTS(v) is a partition of all triples of a v-set into disjoint KTS(v)s — a classical open existence problem since Kirkman's schoolgirl problem of 1850. An overlarge set OLKTS(v) is the variant in which one takes a set $X$ with $|X|=v+1$ and partitions all triples of $X$ into $v+1$ block sets $\mathscr{B}_x$, $x \in X$, such that each $(X \setminus \{x\}, \mathscr{B}_x)$ is a KTS(v). Necessarily $v \equiv 3 \pmod 6$. Known infinite families include the multiplicative families $v=\prod(2q_i^{r_i}+1)\prod(4^{s_j}-1)$ with $q_i \equiv 7 \pmod{12}$ prime powers obtained via the strengthened objects OLKTS$^*$ of Yuan and Kang (2010) — which include OLKTS($3^m$) by taking all factors equal to $4^1-1=3$ — together with OLKTS($2 \cdot 13^n+1$) of Kang and Yuan (2017); but no construction is known for all admissible v. Two auxiliary structures drive the recursive theory. A Kirkman frame KF($m^u$) (Stinson, 1987) is a resolvable group divisible design with u groups of size m and block size 3: every pair of points from distinct groups lies in exactly one block, and the blocks partition into partial parallel classes (holey parallel classes), each of which misses exactly one group. An OLKF($m^u$) (Yuan and Kang, 2009), an overlarge set of Kirkman frames, is a partition of all triples of a set $Y$ of $mu$ points, arranged in $u$ groups of size $m$, that meet three distinct groups, into $mu$ Kirkman frames: the component indexed by a point $a \in Y$ is a KF($m^{u-1}$) on $Y$ with the group containing $a$ deleted, and every triple meeting three distinct groups is a block of exactly one component. Note the type notation: $6^k$ means k groups of size 6, i.e. 6k points, so an OLKF($6^k$) has $6k$ components, each of type $6^{k-1}$. A K-OLGKS($m^u, w$) (Yuan and Kang, 2009), an overlarge set of generalized Kirkman systems, for a set $K$ of positive integers (for a 3-OLGKS, $K=\{3\}$), is a collection $\{(X \setminus \{x\}, \mathscr{B}_x) : x \in Y\} \cup \{(Y, \mathscr{G}, \mathscr{A}_j) : 1 \le j \le w\}$, where $Y$ is a set of $mu$ points partitioned into $u$ groups $\mathscr{G}$ of size $m$, $Y_1$ is a disjoint $w$-set, and $X = Y \cup Y_1$, such that: (i) each $(Y, \mathscr{G}, \mathscr{A}_j)$, $1 \le j \le w$, is a Kirkman frame KF($m^u$); (ii) each $(X \setminus \{x\}, \mathscr{B}_x)$, $x \in Y$, is a modified resolvable system MRS($K \cup \{m+w-1\}, mu+w-1$) — a pairwise design $S(2, K \cup \{m+w-1\}, mu+w-1)$ whose block set is partitioned into parallel classes such that every block $B$ lies in exactly $(|B|-1)/2$ of them — containing the distinguished block $(G \setminus \{x\}) \cup Y_1$ of the group $G$ with $x \in G$; (iii) every triple $T \subseteq X$ with $|T \cap (G \cup Y_1)| \le 2$ for every group $G \in \mathscr{G}$ lies in exactly one block of $(\bigcup_{x \in Y} \mathscr{B}_x) \cup (\bigcup_{j=1}^w \mathscr{A}_j)$; and (iv) every block $B$ occurring in $\bigcup_{x \in Y} \mathscr{B}_x$ with $|B| \in K$ occurs in exactly $|B| - 2$ of the families $\mathscr{B}_x$. The load-bearing composition lemma is: a 3-OLGKS($6^u,4$) together with an OLKTS(9) containing a sub-OLKTS(3) yields an OLKTS($6u+3$). Yuan and Kang (2009) combined this with S(3,K,v) skeletons (3-wise balanced designs with block sizes in $K=\{5,\dots,40\}\setminus\{17,21,22,25,26\}$; the k=5 modules are constructed in the paper itself and the k=8 modules were already available from earlier constructions, which is why Theorem C's hypothesis starts at k=6 and omits k=8, and sizes 17,21,22,25,26 are never used) to prove their Theorem C: if OLKF($6^k$) and 3-OLGKS($6^{k-1},4$) exist for all $k \in \{6,7,\dots,40\}\setminus\{8,17,21,22,25,26\}$, then OLKTS(v) exists for every $v \equiv 3 \pmod 6$, $v \neq 21$. They also proved unconditionally that OLKTS($6u+3$) exists for $u=2^{2n-1}-1$, $7^n$, $31^n$, $127^n$, and $4^r 25^s$ ($n \geq 1$, $r+s \geq 1$). A closely related structure is an RDSQS(n+1), a Steiner quadruple system of order n+1 whose derived design at every point is resolvable: since the derived designs of an SQS(n+1) partition all triples of the point set into STS(n)s, an RDSQS(n+1) is exactly an OLKTS(n); this requires $n+1 \equiv 4 \pmod 6$. Follow-up work has built mainly on the RDSQS/OLKTS$^*$ side: Xu and Ji (2022) produced new infinite families of LR-designs and OLKTSs; Liu and Lei (2023) constructed explicit RDSQSs including RDSQS(22); Xu and Ji (2024) introduced OLKTS$^+$(v), an overlarge-set reformulation of the RDSQS(v+1) condition that plays the same role for LKTS(3v); Tan and Zhou (2025, 2026) added further RDSQS families and proved that OLKTS($3^m a$) exists for every $a \equiv 3 \pmod 6$ with $a < 100$ except $a \in \{51,69,75\}$. None of these works touches the OLKF($6^k$)/3-OLGKS($6^{k-1},4$) checklist, which therefore remains — seventeen years later — the only published finite reduction of the OLKTS existence problem to a finite list of explicitly bounded auxiliary designs.

Problem Statement

For every $k \in \{6,7,\dots,40\}\setminus\{8,17,21,22,25,26\}$ (29 values in total), establish whether there exists an overlarge set of Kirkman frames OLKF($6^k$) — that is, on 6k points arranged in k groups of size 6, the partition of the triples meeting three distinct groups into 6k Kirkman frames of type $6^{k-1}$ — and whether there exists a 3-OLGKS($6^{k-1},4$), both in the precise sense defined by Yuan and Kang (2009). By Theorem C of that paper, affirmative answers for all such k imply that an OLKTS(v) exists for every $v \equiv 3 \pmod 6$ with $v \neq 21$; since OLKTS(21) has now been constructed independently (via the RDSQS(22) of Liu and Lei, 2023), completing this checklist would settle the existence classification of overlarge sets of Kirkman triple systems for all admissible orders with no exception. A proof that an OLKF($6^k$) or a 3-OLGKS($6^{k-1},4$) does not exist for some $k$ in this set would show that Theorem C's route cannot be completed as stated, and would require other machinery for the corresponding orders.

References

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