bibkey: ghafari2026transversals authors: Afsane Ghafari, Ian M. Wanless year: 2026 title: Latin Squares whose transversals intersect in unusual ways doi: 10.48550/arXiv.2607.17547 url: https://arxiv.org/abs/2607.17547v1 claim: Equation (7) defines the Latin square H; Lemma 8 proves pairwise intersection of its transversals for k >= 9, and Section 3 establishes no pinned entry for 9 <= k <= 2500. strata_touched: [] license: citation-only triage: anchor

Ghafari–Wanless H family: source and scope

The primary source is arXiv:2607.17547v1, submitted 20 July 2026. The version-specific TeX source is available from https://arxiv.org/src/2607.17547v1. Its TransAlgo.tex has SHA-256 fbc03bc2e8a4ebf0c0aac4e76c27fc1ead40c81f0dbee379720cfe2fc80d3a3e.

Verified locator

DOI: 10.48550/arXiv.2607.17547

URL: https://arxiv.org/abs/2607.17547v1

• Locator: Section 3, equation (7), source label Structuretwo, defines H of order n = 4k and gives the source’s Latinness assertion; Lemma 5, source label l:Delta, gives the transversal Delta-sum congruence; Lemma 8, source label zeromodfour, proves no pair of disjoint transversals for k >= 9.
• Locator: The paragraph immediately before Section 4 reports three transversals with empty total intersection for the finite range 9 <= k <= 2500. The uniform construction recorded below is separate from that finite report.

The following assertions are literature-attested:

• Section 3, equation (7), source label Structuretwo, defines H of order n = 4k and asserts that it is Latin. Its cases have priority in their printed order. The source initially defines this family for k >= 4; the manuscript uses only k >= 9.
• Lemma 5, source label l:Delta, gives the transversal Delta-sum congruence n/2 modulo n when n is even.
• Lemma 8, source label zeromodfour, proves that H has no pair of disjoint transversals when k >= 9. Its row bounds force every transversal to contain at least two of (1,1,5), (6,5,14), (11,9,23). Although the proof introduces its aim using the words “exactly two”, the concluding argument establishes “at least two”; that is the assertion used in the manuscript.
• The paragraph near the end of Section 3, immediately before Section 4, reports three transversals with empty total intersection for every H of order 4k with 9 <= k <= 2500. These finite cases are prior art. Theorem 6 establishes the larger finite range n = 28 and even 32 <= n <= 10000, using both families and separate small constructions.
• Conjecture 3 asks for a Latin square with pairwise-intersecting transversals and no pinned entry at every even order n >= 28. In particular, its order-30 existence assertion is not among the finite cases of Theorem 6.

The explicit uniform formulas and proofs in LATIN_H_TRANSVERSALS.md are repo-derived ordinary mathematics. Their Latinness premise is the source assertion accompanying equation (7). The construction supplies three transversals for every k >= 9; the source’s finite search result is not used as a premise of that uniform construction. The two current D5 consumers are D5/S3/Combinatorics/Latin/LatinHTransversals.transversal_obstruction and D5/S3/Combinatorics/Latin/LatinHFamilyTheorem.result; the latter keeps Latinness of the actual square as the explicit source premise. Full Conjecture 3 remains unresolved by this H-family result; neither a result for G nor an order-30 square with the conjectured properties follows from it.

As of 21 September 2026, the arXiv version record lists only v1. The bounded arXiv query all:"transversals" AND (au:Ghafari OR au:Wanless) returns 13 entries, with this paper the most recent; OpenAlex work W7169882857 has zero indexed citing works, also zero in its direct citing-work query. No later uniform construction for this same H family was located in those results. These restricted indexes do not establish novelty or exclude other authors, unpublished work, or unindexed sources.