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A Combinatorial Proof of Hilton's Conjecture and Beyond

  • Thomas Lesgourgues
  • Luke Postle
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Abstract

Using refined absorption, we prove that for every integer g≥ 1 and real γ> 0, and for sufficiently large n, there exists an n−1+γ-spread distribution on Latin squares of order n and girth at least g that have no proper subsquares. This implies a combinatorial proof of Hilton's conjecture from the 1970s (recently proved algebraically by Allsop and Wanless) that for all sufficiently large n, there exists a subsquare-free Latin square of order n; indeed, it implies there exist at least n(1−o(1))n2 subsquare-free squares. Simultaneously it also implies the existence of high girth Latin squares (recently proved by Kwan, Sah, Sawhney and Simkin) and even an n−1+γ-spread distribution on high girth Latin squares.

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Cite

BibTeXRIS
BibTeX
@misc{lesgourgues2026combinatorial,
  title = {{A Combinatorial Proof of Hilton's Conjecture and Beyond}},
  author = {Thomas Lesgourgues and Luke Postle},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28966},
  eprint = {2609.28966},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28966},
}
RIS
TY  - UNPB
TI  - A Combinatorial Proof of Hilton's Conjecture and Beyond
AU  - Thomas Lesgourgues
AU  - Luke Postle
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.28966
UR  - https://arxiv.org/abs/2609.28966
AB  - Using refined absorption, we prove that for every integer g≥ 1 and real γ> 0, and for sufficiently large n, there exists an n^−1+γ-spread distribution on Latin squares of order n and girth at least g that have no proper subsquares. This implies a combinatorial proof of Hilton's conjecture from the 1970s (recently proved algebraically by Allsop and Wanless) that for all sufficiently large n, there exists a subsquare-free Latin square of order n; indeed, it implies there exist at least n^(1−o(1))n^2 subsquare-free squares. Simultaneously it also implies the existence of high girth Latin squares (recently proved by Kwan, Sah, Sawhney and Simkin) and even an n^−1+γ-spread distribution on high girth Latin squares.
ER  -