A transversal of a latin square of order nn is a set of nn entries representing every row, column, and symbol exactly once; an entry is pinned if it lies in every transversal of its square. Ghafari and Wanless (arXiv:2412.12466) constructed, for every even n≥10n\ge 10, latin squares whose transversals all coincide on ⌊n/6⌋\lfloor n/6\rfloor entries, and posed a landscape of open problems on how strongly transversals can be forced to coincide. Let p(n)p(n) be the maximum number of pinned entries in an order-nn latin square that has at least one transversal. The open targets are: (i) a nontrivial upper bound on the number of pinned entries in an order-nn square with at least two transversals, asymptotically below the trivial n−3n-3, and constructions with more than ⌊n/6⌋\lfloor n/6\rfloor pinned entries; (ii) whether any latin square of odd order n>9n>9 has a pinned entry at all (pinned entries exist at orders 5 and 7, and none exist at order 9); (iii) whether a latin square of odd order n>9n>9 can have exactly one transversal, the extreme case p(n)=np(n)=n, which parity rules out at even orders; and (iv) for which orders there exists a latin square that has transversals, has no pair of disjoint transversals, and has no pinned entry — a 2026 follow-up by the same authors established such squares for n=28n=28 and all even 32≤n≤10 00032\le n\le 10\,000, and none exists below order 10, leaving the even orders 10≤n≤2610\le n\le 26, n=30n=30, even n>10 000n>10\,000, and all odd orders n>9n>9 open.

Background

A latin square of order nn is an n×nn\times n array of nn symbols in which each symbol appears exactly once in each row and each column; the addition table of a group (its Cayley table) is the canonical example. A transversal is a selection of nn entries of the square that uses each row, each column, and each symbol exactly once. Transversals matter because a decomposition of a latin square into nn pairwise disjoint transversals is exactly what is needed to build an orthogonal mate — a second latin square superimposed on the first so that every ordered pair of symbols occurs exactly once — and because Ryser's conjecture (Ryser, 1951), still open, predicts that every latin square of odd order has a transversal (companion conjectures predict that every latin square has a partial transversal of size n−1n-1).

This problem concerns how constrained the family of transversals of a single square can be. An entry is pinned if it belongs to every transversal (and at least one transversal exists); an entry is transversal-free if it belongs to no transversal; two transversals are disjoint if they share no entries. Two structural facts delimit what is possible. First, a parity theorem of Balasubramanian (1990): a latin square of even order has an even number of transversals, so no even-order square has a unique transversal. Second, Cavenagh and Wanless (2010) showed that two distinct transversals share at most n−3n-3 entries (and that every value up to n−3n-3 is attainable); consequently any square with at least two transversals has at most n−3n-3 pinned entries. No stronger upper bound is known.

On the constructive side, Egan and Wanless (2012) built, for every even n≥10n\ge 10, a latin square with at least one pinned entry, and squares of order n=3mn=3m with mm odd containing an (m−1)×m(m-1)\times m subrectangle consisting entirely of transversal-free entries — still the only known odd-order family with more than a constant fraction of transversal-free entries. Ghafari and Wanless (2024) strengthened both fronts: for every even n≥10n\ge 10 there is a latin square in which all transversals coincide on ⌊n/6⌋\lfloor n/6\rfloor entries, and those squares have at least 19n2/36+O(n)19n^{2}/36+O(n) transversal-free entries, asymptotically more than half the square, even though transversals exist. These constructions, like most in the area, rest on the Delta-lemma: for a square whose rows, columns, and symbols are indexed by the cyclic group Zn\mathbb{Z}_n, the quantity Δ(r,c,s)=s−r−c mod n\Delta(r,c,s)=s-r-c \bmod n sums, over the entries of any transversal, to 00 when nn is odd and to n/2n/2 when nn is even; engineering the Δ\Delta-values of chosen entries forces every transversal through them.

Odd orders behave very differently. Pinned entries exist at orders 5 and 7, complete enumeration shows that none exist at order 9, and for no odd order above 9 is any example known; the Delta-lemma's even-order parity obstruction, which powers the constructions, disappears at odd orders. How abundant transversals are at odd orders is itself open: writing t(n)t(n) for the minimum number of transversals in an order-nn latin square, t(n)=0t(n)=0 for even nn (the Cayley table of Zn\mathbb{Z}_n has no transversal when nn is even, by the Delta-lemma), while for odd nn no nontrivial lower bound is known. Dai, Divoux, and Kelly (2025) recently gave the first significant upper bound, t(n)≤(n/e2.117)nt(n)\le (n/e^{2.117})^{n} for odd nn divisible by 3, and conjectured t(n)≥(n/C)nt(n)\ge (n/C)^{n} for all odd nn, which would rule out a unique transversal; this remains conjectural. For context on typicality: Bowtell and Montgomery (2025) proved that asymptotically almost every latin square decomposes entirely into disjoint transversals, while Cavenagh and Wanless (2017) showed there are superexponentially many transversal-free squares of each even order, so heavily constrained squares are an atypical, even-order-tilted phenomenon. The most recent word is Ghafari and Wanless (2026): for every even n≥28n\ge 28 except n=30n=30 they construct squares with no two disjoint transversals, and for n=28n=28 and all even 32≤n≤10 00032\le n\le 10\,000 they verify computationally that these squares also have no pinned entry — squares whose transversals pairwise intersect yet have no universal common entry.

Problem Statement

Let p(n)p(n) denote the maximum number of pinned entries in a latin square of order nn that has at least one transversal. The problem is to map how strongly the transversals of a latin square can be forced to coincide, in four linked facets:

(1) Extremal bounds on pinning. Determine p(n)p(n), or at minimum prove a nontrivial upper bound on the maximum number of pinned entries over order-nn squares with at least two transversals — any bound asymptotically smaller than the trivial cap of n−3n-3, which holds for every square with at least two transversals, or otherwise materially stronger than this cap — and decide whether p(n)p(n) can exceed ⌊n/6⌋\lfloor n/6\rfloor, the best known lower bound, attained for every even n≥10n\ge 10.

(2) Odd-order existence. Does there exist a latin square of any odd order n>9n>9 containing at least one pinned entry? (Pinned entries exist at orders 5 and 7, and are known not to exist at order 9.)

(3) The unique-transversal extreme. Does there exist a latin square of odd order n>9n>9 with exactly one transversal — equivalently, is p(n)=np(n)=n for some odd n>9n>9? (Even orders are excluded by Balasubramanian's parity theorem; the case of odd orders above 9 is open, though expected to have a negative answer.)

(4) Coincidence without pinning. For which orders nn does there exist a latin square that has at least one transversal, has no pair of disjoint transversals, and has no pinned entry? Such squares are impossible below order 10 and are now known for n=28n=28 and every even 32≤n≤10 00032\le n\le 10\,000; the remaining open cases are the even orders 10≤n≤2610\le n\le 26, n=30n=30, even n>10 000n>10\,000, and all odd orders n>9n>9, and the source conjectures existence for all sufficiently large even orders.

Facets (1)-(3) concern one extremal function p(n)p(n) — its growth, its positivity at odd orders above 9, and whether it attains its maximum possible value nn — while facet (4) probes the boundary of the same phenomenon from the other side: a pinned entry trivially forbids disjoint transversal pairs, and facet (4) asks whether that implication has no converse.

References

  1. Afsane Ghafari and Ian M. Wanless, Latin Squares whose transversals share many entries, arXiv:2412.12466 (2024). DOI: 10.48550/arXiv.2412.12466, https://arxiv.org/abs/2412.12466
  2. Afsane Ghafari and Ian M. Wanless, Latin Squares whose transversals intersect in unusual ways, arXiv:2607.17547 (2026). https://arxiv.org/abs/2607.17547
  3. Judith Egan and Ian M. Wanless, Latin Squares with Restricted Transversals, Journal of Combinatorial Designs 20:344-361 (2012). DOI: 10.1002/jcd.21316, https://doi.org/10.1002/jcd.21316
  4. Nicholas J. Cavenagh and Ian M. Wanless, On the number of transversals in Cayley tables of cyclic groups, Discrete Applied Mathematics 158:136-146 (2010). DOI: 10.1016/j.dam.2009.09.006, https://doi.org/10.1016/j.dam.2009.09.006
  5. K. Balasubramanian, On transversals in latin squares, Linear Algebra and its Applications 131:125-129 (1990). DOI: 10.1016/0024-3795(90)90378-P, https://doi.org/10.1016/0024-3795(90)90378-P
  6. Nicholas J. Cavenagh and Ian M. Wanless, Latin Squares with No Transversals, The Electronic Journal of Combinatorics 24(2):#P2.45 (2017). DOI: 10.37236/6481, https://doi.org/10.37236/6481
  7. H. J. Ryser, A combinatorial theorem with an application to latin rectangles, Proceedings of the American Mathematical Society 2:550-552 (1951). DOI: 10.1090/S0002-9939-1951-0042361-0, https://doi.org/10.1090/S0002-9939-1951-0042361-0
  8. Candida Bowtell and Richard Montgomery, Almost every Latin square has a decomposition into transversals, arXiv:2501.05438 (2025). https://arxiv.org/abs/2501.05438
  9. Tantan Dai, Alexander Divoux, and Tom Kelly, Entropy Bounds for Perfect Matchings in Bipartite Hypergraphs, arXiv:2506.17652 (2025, revised 2026). https://arxiv.org/abs/2506.17652
  10. Billy Child and Ian M. Wanless, Latin hypercubes with restricted transversals, arXiv:2605.01813 (2026). https://arxiv.org/abs/2605.01813