Universal probability bounds for partial Latin squares

Mathematics > Combinatorics [Submitted on 16 Jun 2026] Title:Universal probability bounds for partial Latin squares View PDFAbstract:This paper studies the probability of substructures occurring in random Latin squares. Our main result states that if $\alpha,\beta>0$ are such that $2\alpha+\beta<1$, then there are positive constants $\delta = \delta(\alpha, \beta)$ and $\Delta = \Delta(\alpha, \beta)$ such that if $P$ is a partial Latin square of order $n$ with $k = k(n)$ non-empty cells occupying at most $\alpha n$ rows and $\beta n$ columns, the probability that a random Latin square of order $n$ contains $P$ lies between $(\delta/n)^k$ and $(\Delta/n)^k$. We apply this result to subsquares in random Latin squares to obtain the first proof of the fact that the expected number of subsquares of order $3$ in a random Latin square of order $n$ is non-vanishing as $n \to \infty$. We are also able to provide the best known asymptotics for the expected number of subsquares of order $a$ in a random Latin square of order $n$ when $2<a=o(n^{1/2})$. Finally, we discuss the implications of our result on other configurations in random Latin squares as well as on completions of partial Latin squares. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.