Almost every Latin square has a decomposition into transversals

Candida Bowtell; Richard Montgomery

Almost every Latin square has a decomposition into transversals

T0 review · 0 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A random Latin square of order n decomposes into n transversals with probability 1-o(1).

desk verdict A long, intricate proof that almost every Latin square decomposes into transversals; the central claim looks right and the paper deserves a serious referee. read the letter →

arxiv 2501.05438 v1 pith:CUEC2JNQ submitted 2025-01-09 math.CO

Candida Bowtell , Richard Montgomery This is my paper · ORCID

classification math.CO MSC 05B1505D4005C70

keywords Latinsquarestransversalsdecompositionintorandomrainbowperfectmatchingsabsorptionmethodsemi-randomresolvabledesigns

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper resolves the natural quantitative version of a question first posed in 1782: instead of asking for which orders a Latin square can be decomposed into transversals, it asks how common such squares are. It proves that a uniformly random Latin square of order $n$ can be partitioned into $n$ transversals with probability $1-o(1)$, so the known counterexamples to the classical conjecture are asymptotically rare. In graph form, a random optimal colouring of $K_{n,n}$ has a decomposition into $n$ rainbow perfect matchings with probability $1-o(1)$. This makes the hypergraph of a random Latin square resolvable in the design-theoretic sense, a property previously known only approximately, with nearly $n$ matchings covering nearly all cells.

What carries the argument

The proof runs on three mechanisms that interlock. First, an absorption schematic: a deterministic, random-square-independent collection of pairs $\{(i,u),(j,v)\}$ telling which vertex $u$ from which target matching $i$ should be exchanged with which $v$ from matching $j$, built to handle any balanced degree correction; it reduces the number of required switchers from $\Theta(n^4)$ to $O(n^2)$. Second, the semi-random method for almost-regular hypergraphs with small codegree, which finds the near-matchings and the switching paths, using a pseudorandom matching theorem that controls prescribed weight functions. Third, tight counting of links: for each pair of vertices in the same part, the number of length-62 paths whose odd and even halves use the same 31 distinct colours in the same order is $(1\pm\varepsilon)n^{30}$, with nine constrained variants also tightly bounded; these counts keep the auxiliary hypergraphs near-regular and their codegrees small. The counts come from applying the deletion method both for upper and lower bounds, on top of switching-method probability estimates for embedding small coloured subgraphs.

What would settle it

For a finite check, sample many uniformly random Latin squares of order $n$ (feasible for $n$ up to about 12) and count the special 62-edge switching paths between a fixed pair of same-part vertices: if the count does not concentrate near $n^{30}$, the paper's counting lemma fails. For an asymptotic refutation, exhibit an infinite family of Latin squares that occupies a positive fraction of all order-$n$ squares while lacking a transversal decomposition, which would directly contradict Theorem 1.1.

Watch

Extended reading notes

Core claim

The central discovery is that exact decomposition can be forced by a sparse, pre-built template of local switches, while approximate decomposition is easy. The paper constructs an absorption schematic: for each tribe of target matchings, a set of instructions specifying which vertices must be swapped between which matchings to correct any balanced set of degree errors. It then shows that a random optimal colouring of $K_{n,n}$ contains, with probability $1-o(1)$, the switching paths that realise these instructions: edge-disjoint length-62 paths whose odd and even halves repeat the same 31 colours in order, counted tightly at $(1\pm\varepsilon)n^{30}$ per vertex pair. These paths let the proof upgrade an approximate decomposition into $n$ disjoint rainbow perfect matchings. The net statement is that almost every Latin square—including almost every one of the orders once believed to be impossible—admits an exact decomposition.

Load-bearing premise

The construction depends on an exact-enough census of special 62-edge switching paths between every eligible vertex pair and on nine related constrained counts; if any of these estimates were off by a constant factor, the auxiliary hypergraphs would stop being near-regular and the decomposition would fail.

Editorial extensions

If this is right

For every $n \equiv 2 \pmod{4}$, the proportion of Latin squares of order $n$ that fail to decompose into transversals tends to $0$; the classical counterexample orders are, in the limit, overwhelmingly decomposable.
A uniformly random optimal colouring of $K_{n,n}$ decomposes into $n$ rainbow perfect matchings with probability $1-o(1)$, giving a random-graph form of resolvability.
The 3-uniform hypergraph associated with a random Latin square is resolvable with high probability, placing random Latin squares in the same class as the resolvable designs whose large-$n$ existence is known deterministically.
The absorption schematic is independent of the random square and so is a reusable template for exact decomposition arguments in other sparse structures.
The lower-bound application of the deletion method is new: the paper shows how to certify lower bounds on rare-substructure counts in random Latin squares, not just upper bounds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The natural next target is the analogue for almost all Steiner triple systems: the paper notes that its switching-method inputs have no known non-bipartite counterpart, so that extension would need a new route to the required substructure counts.
The sharp concentration of link counts around $n^{30}$ suggests that a second-moment argument might give quantitative estimates on the fraction of decomposable squares, strengthening the current $1-o(1)$ to an explicit error bound.
A testable refinement is that the same proof should cover random $k \times n$ Latin rectangles with $k$ close to $n$, since all probability inputs are modular with respect to the colour set.
The absorption schematic itself is deterministic; in graphs that already contain the right density of colour-repeating paths, the same template could certify exact rainbow decompositions without any randomness.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know about this paper before it gets lost in the queue: it resolves van Rees's 1990 conjecture by showing a random Latin square of order n has a transversal decomposition with probability 1-o(1). That is a real theorem, not a heuristic. What is genuinely new: prior work only guaranteed a single transversal with high probability (Kwan 2020) or almost-resolvability (Ferber-Kwan). The paper builds an absorption schematic that is independent of the random Latin square, then realizes it via three applications of the semi-random method. The theorem also confirms the Wanless-Webb numerics and shows that, for n equiv 2 mod 4, Euler's conjecture fails in the typical case, not just by sparse explicit constructions.

The proof is long, roughly 90 pages, but carefully structured. I read the division into Parts A, B, C and the key Lemmas 3.4 and 3.5: the architecture is coherent. Part A constructs the schematic, Part B finds edge-disjoint rainbow near-matchings realizing it, Part C covers and balances the leftovers. The counting in Section 5 is the engine room, and the most fragile point is Theorem 5.2, especially the lower bound in Lemma 5.12, which uses a novel deletion-method lower bound with a conditioning step at equation (26). I checked the stress-test note's algebra there: the conditioning looks sound, the constant cancels in the ratio, and there is no circular dependency between K4 and K1-K3. So I have no concrete objection to the central claim.

Soft spots, in proportion: the paper relies heavily on prior results, some co-authored by Montgomery, but those are published theorems with independent proofs, so that is not circular. The real risk is that a 90-page proof with this much concentration machinery can hide a small gap, and a referee will need to spend serious time on Section 5 and on the hypergraph applications in Parts B and C. That is the price of the method, not a flaw I can put my finger on.

Who benefits: specialists in probabilistic combinatorics, Latin squares, random designs, and absorption. The absorption schematic itself may be reusable elsewhere. This paper deserves a serious referee; if the long counting arguments hold up, it is a major result. I would engage with it and would cite it in my own work.

Referee Report

0 major / 3 minor

Summary. The paper proves Theorem 1.1: a uniformly random Latin square of order n has, with probability 1-o(1), a decomposition into n disjoint transversals. This is formulated equivalently in Theorem 2.1 for a uniformly random optimal colouring of K_{n,n}. The proof is modular: Section 3 reduces the theorem to three key lemmas; Part A (Section 4) constructs a sparse absorption schematic; Part B (Section 6) realizes the schematic in the random colouring, using the path-counting estimates of Theorem 5.2; Part C (Section 7) covers, balances, and partitions the remaining edges. The final combination is carried out in Section 3.6.

Significance. If correct, the result settles the probabilistic counterpart of Euler's 1782 conjecture: not only do counterexamples to Euler's conjecture exist for every admissible order, but they have density o(1) among Latin squares. It confirms the numerical prediction of Wanless and Webb against van Rees's conjecture, and extends Kwan's earlier result on a single transversal to full resolvability of the associated 3-partite 3-uniform hypergraph. The paper also makes methodological contributions: the absorption schematic in Part A is an explicit and reusable template, and Section 5 develops deletion-method lower bounds for counts of rare structures in random Latin squares, a step that had previously been approached with switching methods. The overall architecture is clear and the lemmas are stated in a checkable way.

minor comments (3)

[Section 3.1, equations (2)-(5)] The derivation of p_pt = (1 ± sqrt(beta)) alpha p_T is correct, but it would aid the reader if the text stated explicitly that p_W is the quantity eliminated from p_S + p_X + p_Y + p_Z = 1 after substituting p_S = p_U + p_V + p_W. As written, the reader has to reverse-engineer which variable is being solved for.
[Section 5.1, first paragraph of the deletion-method sketch] The phrase 'using Corollary 2.5 is far off even the trivial bound P(H subset G) <= 1' is grammatically awkward and slightly ambiguous; rephrasing as 'gives a bound that is far weaker than the trivial bound' would improve readability.
[Section 5.6, proof of K4] The counting in the cases 16 <= k <= 31 and 2 <= k <= 15 relies on a decomposition of the L-link into segments P_i and Q_i that is defined only informally in the text. A sentence stating the segment lengths explicitly (24, 7, and 31 edges, in the cases used) would make the verification substantially easier.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central absorption construction is self-contained, and the external citations are independent published results, not load-bearing self-citations.

full rationale

The paper's central claim is an absorption-based construction: Part A builds a purely combinatorial absorption schematic from random vertex partitions (Lemma 3.3), Part B realises it using counts of short L-links (Theorem 5.2), and Part C completes the matchings via the semi-random method and a final edge partition. The counting theorem, Theorem 5.2, is proved inside the paper using the deletion method and the switching-method estimate Corollary 2.5, which is itself derived from external results of McKay-Wanless and Godsil-McKay via Kwan-Sah-Sawhney. These external results do not assume the target statement, and the deletion-method lower bounds are developed from scratch in Section 5. The absorption schematic in Part A is independent of the random Latin square and does not import the conclusion. The variable consistency check at (3)-(5) is an internal algebraic verification, not an equation equating the prediction to an input. Lemma 3.4's condition B4 is an implication, and Lemma 3.5 supplies the required matchings; this is a normal proof structure, not circularity. The self-citations (e.g., Montgomery-Pokrovskiy-Sudakov [31], Montgomery survey [30]) appear in the introduction and proof overview as context or motivation, not as load-bearing ingredients; the actual hypergraph matching theorem used is the independent Ehard-Glock-Joos result. No equation in the paper reduces by construction to its own input, and no fitted parameter is renamed as a prediction. The honest finding is no significant circularity, with at most minor, non-load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No data are fitted. The many small probability parameters (ptr, pfa, epsilon, gamma, beta, pcov, pbal, ppt, pT, pU, pV, pW) are analytical constants determined by the hierarchy at (2); none are estimated from data or from the target statement. The axioms listed are the external theorems and standard inequalities the proof relies on. No new physical or mathematical entities are postulated.

assumptions (5)

standard math Chernoff and McDiarmid concentration inequalities (Lemmas 2.6 and 2.7)
Used throughout to control deviations of random counts and partition sizes.
standard math Theorem 2.2: semi-random method for hypergraph matchings (Ehard-Glock-Joos)
Provides almost-perfect pseudorandom matchings in auxiliary hypergraphs, used in Parts B and C.
domain assumption Theorems 2.3 and 2.4: switching-method estimates for random Latin squares (McKay-Wanless, Godsil-McKay via Kwan-Sah-Sawhney)
External estimates on probabilities of small subgraphs, converted into Corollary 2.5 and used in all deletion-method arguments in Section 5.
domain assumption Uniform random optimal colouring of K_{n,n} is equivalent to uniformly random Latin square
Standard bijection described in Section 2.1; every properly coloured K_{n,n} with n colours corresponds to a Latin square.
standard math Tutte's theorem for perfect matchings in auxiliary graphs
Used in Section 4.7 to show the existence of a perfect matching in the auxiliary graph L_A after random sparsification.

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Pith. "Pith review of Almost every Latin square has a decomposition into transversals." pith.science (2026). https://pith.science/paper/CUEC2JNQ

@misc{pith2026250105438,
  author       = {Pith},
  title        = {Pith review of: Almost every Latin square has a decomposition into transversals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CUEC2JNQ}},
  note         = {Machine review of arXiv:2501.05438}
}

abstract

In 1782, Euler conjectured that no Latin square of order $n\equiv 2\; \textrm{mod}\; 4$ has a decomposition into transversals. While confirmed for $n=6$ by Tarry in 1900, Bose, Parker, and Shrikhande constructed counterexamples in 1960 for each $n\equiv 2\; \textrm{mod}\; 4$ with $n\geq 10$. We show that, in fact, counterexamples are extremely common, by showing that if a Latin square of order $n$ is chosen uniformly at random then with high probability it has a decomposition into transversals.

Figures

Figures reproduced from arXiv: 2501.05438 by the authors.

**Figure 2.** A simple {(i, x),(j, y)}-switcher. consisting of edges e1e2 . . . e2s for some s ∈ N such that Modd := {e2k−1 : k ∈ [s]} ⊆ Mi and Meven := {e2k : k ∈ [s]} ⊆ Mj , with Modd and Meven being rainbow matchings with the same colour set. As described above, such an x, y-path P would enable us to switch edges in Mi and Mj such that the updated subgraphs are still rainbow in the same colour set and now in Mi vertex x has de… view at source ↗

**Figure 3.** For each i ∈ [n], we have a partition of A ∪ B into Si ∪ Xi ∪ Yi ∪ Zi (which is the same for individuals i in the same tribe), a partition Si = Ui ∪ Vi ∪ Wi (which is the same for individuals i in the same family) and disjoint sets Ri , Ti ⊂ Ui (which are distinct for each individual i ∈ [n]). 3.2 Tribes, families, and partitions of the vertices, colours and edges We now partition our target matchings into families,… view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: a) Each cycle Cj we consider must be rainbow, for otherwise we would replace, for example, the orange edges 1⃗u u2 and 3⃗u u4 with orange edges 1⃗u u4 and 3⃗u u2. . b) For each rainbow cycle Cj , we take a set Ej = {v4v1, v1v3, v3v2} of edges whose addition allows an (…

**Figure 5.** Figure 5: Each directed triangle xjyj zj is replaced by a collection of 2-cycles, where each vertex has balanced in- and out-degree in each colour except for xj , yj , zj which maintain the same in- and outdegree in each colour. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

**Figure 6.** Figure 6: A u, v-path Pe as found in the sparse auxiliary graph Kϕ using the two binary trees Fwu,re and Fwv,re . In the proof of Lemma 4.4, this path will additionally avoid some set of vertices V forb . case for u, v ∈ B follows similarly. We will find a path Pe, as depicted i…

**Figure 7.** Figure 7: Replacing arrows representing a pair {(i, u),(j, v)} with a sequence of pairs which have the same effect, but take the form {(i ′ , u),(j ′ , v)} for only certain pairs (i ′ , j′ ), as at (8). each ϕ ∈ F, S i∈Iϕ R ′ i =mult S i∈Iϕ Ti, and decompose the corresponding co…

**Figure 8.** Figure 8: The more general structure counted in the proof of Corollary 5.9, where [PITH_FULL_IMAGE:figures/full_fig_p041_8.png]

**Figure 9.** Figure 9: In Part B.2, as on the left, for each i ∈ [n] and u ∈ Si\Ri we find an edge from u to Yi along with a monochromatic matching with the same colour, with one edge for each j such that {(i, u),(j, v)} ∈ I (depicted for {(i, u),(j1, v1)}, {(i, u),(j2, v2)}, {(i, u),(j3, v3…

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Latin Squares whose transversals intersect in unusual ways
math.CO 2026-07 conditional novelty 7.0 of 10

Latin squares of all even orders >= 28 except 30 are constructed so that every two transversals meet while no entry lies in all transversals (proved for orders up to 10,000); dominant transversals exist for all orders...

Reference graph

Works this paper leans on

47 extracted references · 42 canonical work pages · cited by 1 Pith paper

[1]

Alon, J.-H

N. Alon, J.-H. Kim, and J. Spencer. Nearly perfect matchings in regular simple hypergraphs. Israel Journal of Mathematics , 100:171–187, 1997

work page 1997
[2]

Alon and J

N. Alon and J. H. Spencer. The Probabilistic Method . John Wiley & Sons, 2016

2016
[3]

Alon and R

N. Alon and R. Yuster. On a hypergraph matching problem. Graphs and Combinatorics, 21:377– 384, 2005

work page 2005
[4]

L. D. Andersen. The history of Latin squares. In Combinatorics: Ancient & Modern (edited by R. Wilson and J.J. Watkins) , 2013

work page 2013
[5]

Barber, D

B. Barber, D. K¨ uhn, A. Lo, and D. Osthus. Edge-decompositions of graphs with high minimum degree. Advances in Mathematics , 288:337–385, 2016

work page 2016
[6]

R. C. Bose, E. T. Parker, and S. S. Shrikhande. Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler’s conjecture. Canadian Journal of Mathematics , 12:189–203, 1960

work page 1960
[7]

R. C. Bose and S. S. Shrikhande. On the falsity of Euler’s conjecture about the non-existence of two orthogonal Latin squares of order 4 t + 2. Proceedings of the National Academy of Sciences , 45(5):734–737, 1959

work page 1959
[8]

R. A. Brualdi and H. J. Ryser. Combinatorial matrix theory . Cambridge University Press, 1991

work page 1991

Show all 47 references

[9]

Eberhard, F

S. Eberhard, F. Manners, and R. Mrazovi´ c. Transversals in quasirandom Latin squares. Proceed- ings of the London Mathematical Society , 2023

2023
[10]

Ehard, S

S. Ehard, S. Glock, and F. Joos. Pseudorandom hypergraph matchings. Combinatorics, Proba- bility and Computing , 29(6):868–885, 2020

2020
[11]

L. Euler. Recherches sur un nouvelle esp´ ece de quarr´ es magiques.Verhandelingen uitgegeven door het zeeuwsch Genootschap der Wetenschappen te Vlissingen , pages 85–239, 1782
[12]

Ferber and M

A. Ferber and M. Kwan. Almost all Steiner triple systems are almost resolvable. Forum of Mathematics, Sigma , 8:39, 2020

2020
[13]

Frankl and V

P. Frankl and V. R¨ odl. Near perfect coverings in graphs and hypergraphs. European Journal of Combinatorics, 6(4):317–326, 1985. 91

1985
[14]

C. D. Godsil and B. D. McKay. Asymptotic enumeration of Latin rectangles. Journal of Combi- natorial Theory, Series B , 48(1):19–44, 1990

1990
[15]

Gould and T

S. Gould and T. Kelly. Hamilton transversals in random Latin squares. Random Structures & Algorithms, 62(2):450–478, 2023

2023
[16]

Gould, T

S. Gould, T. Kelly, D. K¨ uhn, and D. Osthus. Almost all optimally coloured complete graphs contain a rainbow Hamilton path. Journal of Combinatorial Theory, Series B , 156:57–100, 2022

2022
[17]

Hatami and P

P. Hatami and P. W. Shor. A lower bound for the length of a partial transversal in a Latin square. Journal of Combinatorial Theory, Series A , 115(7):1103–1113, 2008

2008
[18]

P. Keevash. The existence of designs. arXiv preprint arXiv:1401.3665 , 2014

2014 arXiv
[19]

P. Keevash. The existence of designs II. arXiv preprint arXiv:1802.05900 , 2018

2018 arXiv
[20]

Keevash, A

P. Keevash, A. Pokrovskiy, B. Sudakov, and L. Yepremyan. New bounds for Ryser’s conjecture and related problems. arXiv preprint arXiv:2005.00526 , 2020

2005 arXiv
[21]

J. Kim, D. K¨ uhn, A. Kupavskii, and D. Osthus. Rainbow structures in locally bounded colorings of graphs. Random Structures & Algorithms , 56(4):1171–1204, 2020

2020
[22]

T. P. Kirkman. Note on an unanswered prize question. Cambridge and Dublin Math. J , 5:255–262, 1850
[23]

A. V. Kostochka and V. R¨ odl. Partial steiner systems and matchings in hypergraphs. Random Structures & Algorithms , 13(3-4):335–347, 1998

1998
[24]

M. Kwan. Almost all Steiner triple systems have perfect matchings. Proceedings of the London Mathematical Society, 121(6):1468–1495, 2020

2020
[25]

M. Kwan, A. Sah, and M. Sawhney. Large deviations in random Latin squares. Bulletin of the London Mathematical Society, 54(4):1420–1438, 2022

2022
[26]

McDiarmid et al

C. McDiarmid et al. On the method of bounded differences. Surveys in combinatorics, 141(1):148– 188, 1989

1989
[27]

B. D. McKay and I. M. Wanless. Most Latin squares have many subsquares. Journal of Combi- natorial Theory, Series A , 86(2):323–347, 1999

1999
[28]

Montgomery

R. Montgomery. Spanning trees in random graphs. Advances in Mathematics , 356:106793, 2019

2019
[29]

Montgomery

R. Montgomery. A proof of the Ryser-Brualdi-Stein conjecture for large even n. arXiv preprint arXiv:2310.19779, 2023

2023 arXiv
[30]

Montgomery

R. Montgomery. Transversals in Latin squares. Surveys in Combinatorics , 2024

2024
[31]

Montgomery, A

R. Montgomery, A. Pokrovskiy, and B. Sudakov. Decompositions into spanning rainbow struc- tures. Proceedings of the London Mathematical Society , 119(4):899–959, 2019

2019
[32]

J. Ozanam. R´ ecr´ eations math´ ematiques et physiques, volume 1. 1723
[33]

Pippenger and J

N. Pippenger and J. Spencer. Asymptotic behavior of the chromatic index for hypergraphs. Journal of combinatorial theory, Series A , 51(1):24–42, 1989

1989
[34]

Pokrovskiy

A. Pokrovskiy. Rainbow Subgraphs and their Applications , page 191–214. London Mathematical Society Lecture Note Series. Cambridge University Press, 2022

2022
[35]

Ray-Chaudhuri and R

D. Ray-Chaudhuri and R. M. Wilson. Solution of Kirkman’s schoolgirl problem. In Proc. symp. pure Math, volume 19, pages 187–203, 1971

1971
[36]

Ray-Chaudhuri and R

D. Ray-Chaudhuri and R. M. Wilson. The existence of resolvable block designs. In A survey of Combinatorial Theory, pages 361–375. 1973

1973
[37]

V. R¨ odl. On a packing and covering problem. European Journal of Combinatorics , 6(1):69–78, 1985

1985
[38]

R¨ odl and A

V. R¨ odl and A. Ruci´ nski. Threshold functions for Ramsey properties. Journal of the American Mathematical Society, 8(4):917–942, 1995

1995
[39]

R¨ odl, A

V. R¨ odl, A. Ruci´ nski, and E. Szemer´ edi. A Dirac-type theorem for 3-uniform hypergraphs. Combinatorics, Probability and Computing , 15(1-2):229–251, 2006

2006
[40]

H. Ryser. Neuere Probleme der Kombinatorik. Vortr¨ age ¨ uber Kombinatorik, Oberwolfach, pages 69–91, 1967. 92

1967
[41]

P. W. Shor. A lower bound for the length of a partial transversal in a Latin square. Journal of Combinatorial Theory, Series A , 33(1):1–8, 1982

1982
[42]

S. K. Stein. Transversals of Latin squares and their generalizations. Pacific J. Math. , 59:567–575, 1975

1975
[43]

G. Tarry. Le probl´ eme des 36 officiers. Secr´ etariat de l’Association fran¸ caise pour l’avancement des sciences, 1900

1900
[44]

van Rees

G. van Rees. Subsquares and transversals in Latin squares. Ars Combinatoria, 29:193–204, 1990

1990
[45]

I. M. Wanless. Transversals in Latin squares: a survey. Surveys in combinatorics , 392:403–437, 2011

2011
[46]

I. M. Wanless and B. S. Webb. The existence of Latin squares without orthogonal mates. Designs, Codes and Cryptography, 40:131–135, 2006

2006
[47]

R. Wilson. The early history of block designs. Rend. del Sem. Mat. di Messina , (9):267–276, 2003. 93

2003

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Alon, J.-H

N. Alon, J.-H. Kim, and J. Spencer. Nearly perfect matchings in regular simple hypergraphs. Israel Journal of Mathematics , 100:171–187, 1997

work page 1997

[2] [2]

Alon and J

N. Alon and J. H. Spencer. The Probabilistic Method . John Wiley & Sons, 2016

2016

[3] [3]

Alon and R

N. Alon and R. Yuster. On a hypergraph matching problem. Graphs and Combinatorics, 21:377– 384, 2005

work page 2005

[4] [4]

L. D. Andersen. The history of Latin squares. In Combinatorics: Ancient & Modern (edited by R. Wilson and J.J. Watkins) , 2013

work page 2013

[5] [5]

Barber, D

B. Barber, D. K¨ uhn, A. Lo, and D. Osthus. Edge-decompositions of graphs with high minimum degree. Advances in Mathematics , 288:337–385, 2016

work page 2016

[6] [6]

R. C. Bose, E. T. Parker, and S. S. Shrikhande. Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler’s conjecture. Canadian Journal of Mathematics , 12:189–203, 1960

work page 1960

[7] [7]

R. C. Bose and S. S. Shrikhande. On the falsity of Euler’s conjecture about the non-existence of two orthogonal Latin squares of order 4 t + 2. Proceedings of the National Academy of Sciences , 45(5):734–737, 1959

work page 1959

[8] [8]

R. A. Brualdi and H. J. Ryser. Combinatorial matrix theory . Cambridge University Press, 1991

work page 1991