REVIEW 2 major objections 4 minor 1 cited by
Latin Squares whose transversals share many entries
T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read The paper constructs latin squares whose transversals all share n/6 entries.
desk verdict Solid constructive advance: floor(n/6) pinned entries and more than half transversal-free entries for infinitely many even orders; the inclusion-exclusion is intricate but checks out aside from a harmless small-k blemish in Lemma 2.8. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central machinery is the integer-valued function $\Delta(r,c,s) \equiv s-r-c \pmod n$, together with the $\Delta$ Lemma: the sum of $\Delta$ over any transversal is $0$ modulo $n$ when $n$ is odd and $n/2$ modulo $n$ when $n$ is even. For even $n$, a suitable diagonal is any set of $n$ entries, one from each row and column, whose $\Delta$-sum is $n/2$; since every transversal is a suitable diagonal, forcing an entry into every suitable diagonal forces it into every transversal. The construction makes the row-wise maxima of $\Delta$ strict and makes their total exactly $n/2$, so every suitable diagonal must take the maximum from every row, hitting the listed pinned entries. The transversal-free count then comes from inclusion-exclusion over the columns and symbols that the proof forces to avoid all suitable diagonals.
What would settle it
Take the smallest constructed representatives of the three residue classes, $T_{12}$, $V_{16}$, and $U_{20}$, and exhaustively enumerate all suitable diagonals or all transversals. The table reports exact transversal-free counts 67, 107, and 190; if any claimed pinned entry is missing from a suitable diagonal, or if the true count is lower than reported, the inclusion-exclusion lemmas behind Theorem 1.2 would be wrong.
Extended reading notes
Core claim
For orders $n\equiv 0,2,4 \pmod 6$, the paper writes down explicit cyclic latin squares $T_n$, $U_n$, and $V_n$. On each square it defines $\Delta(r,c,s)$ and observes that every transversal must be a suitable diagonal. The squares are engineered so that the sum of the maximum $\Delta$-values over all rows is exactly $n/2$; a suitable diagonal must therefore take the maximum from every row, and the entries listed in (3), (5), and (7) are exactly those maxima. Those $\lfloor n/6 \rfloor$ entries are pinned. Explicit transversals are listed, so the squares are not transversal-free. Inclusion-exclusion over carefully chosen columns and symbols then certifies at least $19n^2/36+O(n)$ entries that no transversal can use. For odd $n=3m$, a separate block construction with $\Delta_m(r,c,s)\equiv r+c \pmod m$ forces every transversal to hit all nine $m\times m$ subsquares.
Load-bearing premise
The proof depends on a long count of which cells are forced to be transversal-free, and that count must be complete; if overlooked exceptions change the totals by more than a constant, the claim that at least $19n^2/36$ cells are transversal-free fails.
Editorial extensions
If this is right
- For every even $n\ge 10$, there is a latin square with at least one transversal but with at least $\lfloor n/6 \rfloor$ entries common to all transversals.
- Those squares have at least $19n^2/36+O(n)$ transversal-free entries, so for $n\ge 88$ more than half the cells are outside every transversal.
- For every odd $m\ge 3$, the order-$3m$ block square has every transversal meeting each of the nine $m\times m$ subsquares.
- The construction improves the previously known one-pinned-entry result for even orders to a pinned set whose size grows linearly with $n$.
- Combined with the parity of the number of transversals, the examples show that the pinned-entry number can be forced to grow while remaining well below the $n-3$ ceiling.
Reading between the lines
- The authors report that their bounds are within $O(n)$ of the true transversal-free counts; if that belief holds, the three constructed families have transversal-free density tending to exactly $19/36$, an asymptotic statement not asserted in the theorems.
- The row-maximum forcing scheme is modular: any even-order square whose row-wise $\Delta$-maxima are strict and sum to $n/2$ produces pinned entries, so perturbing the construction might push the pinned count above $\lfloor n/6 \rfloor$ while staying below the $n-3$ ceiling.
- The odd-order block result is one step short of pinned entries: refining the block-hitting invariant to force a specific cell in each block would yield the first pinned entry for a large odd order, a case where none is currently known.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper constructs, for every even n ≥ 10, a latin square with at least floor(n/6) pinned entries, and at least (19/36)n² + O(n) entries that lie in no transversal. For every odd m ≥ 3 it constructs an order 3m latin square divided into nine m × m subsquares such that every transversal hits each subsquare. The even-order proofs split into the congruence classes n ≡ 0, 2, 4 mod 6, using the Δ-function sum invariant (Lemma 2.1) to force the maximum-Δ entries of selected rows into every suitable diagonal, and then exhibiting explicit transversals. The transversal-free count is obtained by inclusion-exclusion over the rows missing their maximum Δ, together with carefully chosen column and symbol sets N and O. Section 3 introduces a block construction and uses a mod m analogue of the Δ-sum plus automorphisms to prove that every transversal meets every block.
Significance. If the constructions are valid, the results are a clear improvement over the earlier one-pinned-entry theorem of Egan and Wanless and over previous bounds on transversal-free entries in squares that have transversals. The proofs are constructive and elementary, and the claimed numbers are concrete and falsifiable, with Table 1 providing actual counts for small orders that confirm the stated lower bounds. The main theorems are built on the simple Δ-sum lemma, which makes the paper accessible; the bookkeeping in Lemmas 2.6–2.8 is intricate but, as far as I could verify by spot checks, consistent. The odd-order block result is novel and its use of automorphisms to propagate the hitting property is elegant.
major comments (2)
- [§2, definitions (2), (4), (6); §3, definition of L_n] The arrays T_n, U_n, V_n, and L_n are called latin squares, but the paper never proves that each row and column contains every symbol exactly once. This is load-bearing: Lemma 2.6 explicitly invokes the "latin property" to compute |N ∩ O|, and Theorem 1.2 is about latin squares. Please add a verification (or a precise reference) that each of these arrays is a latin square. If the verification is intended to be routine, it should at least be stated explicitly and sketched, because the piecewise definitions involve many exceptional cases.
- [§2, Lemmas 2.6–2.8] The lower bound in Theorem 1.2 rests on the assertion that the exceptional-entry lists (9)–(15) are exhaustive and on the stated sizes of the triple intersections P ∩ N ∩ O, Q ∩ N ∩ O, and R ∩ N ∩ O. The text often says "the structure reveals" or "one can see" for these counts. I did not find an actual error in representative cases, but given how intricate the definitions are, the proof should either expand these derivations into a verifiable case analysis or supply an independent check (for example, a short computer verification for each congruence class) so that completeness is not left to the reader.
minor comments (4)
- [§2, Lemma 2.8, Subcases 3.b and 3.c] The displayed equalities |Q ∩ N ∩ O| = floor((k−3)/2) and |R ∩ N ∩ O| = k−4 are false for small k: for k = 2 they give −1 and −2, which are impossible for cardinalities. Since the proof only uses the subsequent lower bounds (k−4)/2 and k−4, the final bound is unaffected, but the equalities should be stated as lower bounds, or restricted to k ≥ 4 with the small cases handled separately.
- [§2, Lemma 2.8] In the displayed 8 × 8 square, the first row is broken across two lines in the typeset version, making it look like a row of length 4. Please format the full matrix on a single line or in a way that each row is visibly one of the eight rows.
- [§3, equation (16)] The phrase "by a similar argument to Lemma 2.1" is a little terse for the mod m function Δ_m, because Lemma 2.1 concerns Δ modulo n and gives n/2 for even n, whereas here the right-hand side is 0 modulo m. A one-sentence derivation of ∑(r+c) = n²(n−1) ≡ 0 (mod m) would make the step fully explicit.
- [§1, Theorem 1.2] The condition n² > τ(L) is equivalent to the existence of at least one transversal; since this is not stated explicitly, a brief parenthetical remark would help readers who might otherwise wonder why the strict inequality appears alongside the lower bound.
Circularity Check
No circularity found: the construction and counting arguments are self-contained; the only self-citation is an elementary, independently published lemma that does not assume the target results.
full rationale
The paper's main theorems are proven by explicit constructions of latin squares T_n, U_n, V_n and L_n, followed by direct counting arguments. The pinned-entry claim (Theorem 2.2) rests on Lemmas 2.3–2.5, which show that every suitable diagonal must include certain row-maximum entries; this is a direct consequence of the definition of suitable diagonal and the Delta-sum identity, not an assumption of the conclusion. Lemma 2.1, cited from [6] and [7], is a standard modular identity about transversals; it is independently published in [6] by Evans and in [7] by Wanless and Webb, and it is used only as a preliminary tool. Even though one of the citations is coauthored by Wanless, the lemma is elementary, parameter-free, and does not contain the paper's results, so it does not constitute load-bearing self-citation. The transversal-free counting in Lemmas 2.6–2.8 uses inclusion–exclusion with explicit exceptional-entry lists and set sizes; no fitted constants are smuggled in and the target lower bounds are not assumed. Table 1 and the small-case checks support rather than substitute for the proofs. Theorem 1.3 is similarly constructive, using an explicit automorphism/autotopism argument and an explicit transversal. There is no step in which an output quantity is defined in terms of itself, no fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work. The analysis is therefore self-contained apart from the elementary cited lemma, which is legitimate external support.
Assumptions & free parameters
assumptions (3)
- standard math Lemma 2.1 (Delta-sum identity): for any transversal T of a latin square of order n indexed by Z_n, the sum of Delta values is 0 mod n for odd n and n/2 mod n for even n.
- standard math Equation (16): the sum of Delta_m values over any transversal of the odd-order block square is 0 mod m, by the same residue argument as Lemma 2.1.
- standard math Inclusion-exclusion and the latin property are used for the counts in Lemmas 2.6 to 2.8.
Cite this review
Pith. "Pith review of Latin Squares whose transversals share many entries." pith.science (2026). https://pith.science/paper/HACLFPMP
@misc{pith2026241212466,
author = {Pith},
title = {Pith review of: Latin Squares whose transversals share many entries},
year = {2026},
howpublished = {\url{https://pith.science/paper/HACLFPMP}},
note = {Machine review of arXiv:2412.12466}
}
abstract
We prove that, for all even $n\geq10$, there exists a latin square of order $n$ with at least one transversal, yet all transversals coincide on $ \big\lfloor n/6 \big\rfloor$ entries. These latin squares have at least $ 19 n^2/36 + O(n)$ transversal-free entries. We also prove that for all odd $m\geq 3$, there exists a latin square of order $n=3m$ divided into nine $m\times m$ subsquares, where every transversal hits each of these subsquares at least once.
Forward citations
Cited by 1 Pith paper
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Latin Squares whose transversals intersect in unusual ways
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
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Balasubramanian, On transversals in latin squares, Linear Algebra Appl
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[2]
N. J. Cavenagh and I. M. Wanless, On the number of transver sals in Cayley tables of cyclic groups, Disc. Appl. Math. 158 (2010), 136–146
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N. J. Cavenagh and I. M. Wanless. Latin squares with no tra nsversals. The Electronic Journal of Combinatorics , 24(2):#P2.45, 2017
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[4]
J. Egan and I. M. Wanless. Latin squares with restricted t ransversals. Journal of Combinatorial Designs , 20:344–361, 2012
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[6]
A. B. Evans. Latin squares without orthogonal mates. Designs, Codes and Cryptog- raphy, 40:121–130, 2006
work page 2006
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[7]
I. M. Wanless and B. S. Webb, The existence of latin square s without orthogonal mates, Des. Codes Cryptogr. , 40 (2006), 131–135
work page 2006
Reviewed August 11, 2026 · model on record in the stance chip above.
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