We develop a limit theory of Latin squares, paralleling the recent limit theories ofdense graphs and permutations. We introduce a notion of density, an appropriate version ofthe cut distance, and a space of limit objects — so-called Latinons. Key results of our theoryare the compactness of the limit space and the equivalence of the topologies induced by thecut distance and the left-convergence. Last, using Keevash’s recent results on combinatorialdesigns, we prove that each Latinon can be approximated by a finite Latin square.
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