[Paper Review] Limits of Latin squares

Frederik Garbe, Robert Hancock+2 more|arXiv (Cornell University)|Oct 15, 2020

Limits and Structures in Graph Theory43 references4 citations

TL;DR

This paper establishes a limit theory for Latin squares analogous to graphons and permutons, introducing 'Latinons' as analytic limit objects and proving compactness and equivalence of convergence notions via cut distance and left-convergence. It further shows that every Latinon can be approximated by finite Latin squares using Keevash's design theory.

ABSTRACT

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

Motivation & Objective

• To develop a comprehensive limit theory for Latin squares, mirroring existing theories for graphs and permutations.
• To define a notion of density and a cut distance for Latin squares, enabling convergence analysis.
• To introduce 'Latinons' as the analytic limit objects in this theory, analogous to graphons and permutons.
• To prove compactness of the limit space and equivalence between left-convergence and cut-distance convergence.
• To establish that every Latinon can be approximated by finite Latin squares, leveraging combinatorial design theory.

Proposed method

• Define Latinons as measurable functions on the unit cube satisfying threefold symmetry and marginal constraints, ensuring they represent limiting distributions of Latin squares.
• Introduce a cut distance for Latin squares based on the $L^1$-norm of differences in densities over measurable sets, generalizing the graphon cut distance.
• Establish a sampling lemma showing that random Latin squares drawn from a Latinon converge almost surely to the original Latinon.
• Prove the counting lemma, which ensures that densities of substructures in a Latin square converge to those in the corresponding Latinon.
• Use the method of regularity and approximation via hypergraph designs (from Keevash) to prove compactness of the limit space.
• Apply a variant of the hypergraph regularity method and a theorem of Alon and Yuster on perfect matchings in hypergraphs to construct finite approximations of Latinons.

Experimental results

Research questions

• RQ1 Can a limit theory for Latin squares be developed that parallels the theories of dense graphs and permutations?
• RQ2 What are the appropriate analytic limit objects (analogous to graphons and permutons) for sequences of Latin squares?
• RQ3 Is the space of limit objects (Latinons) compact under the cut distance topology?
• RQ4 Are left-convergence and cut-distance convergence equivalent in the context of Latin squares?
• RQ5 Can every Latinon be approximated by a finite Latin square, and if so, under what conditions?

Key findings

• The space of Latinons is compact under the cut distance topology, ensuring every sequence of Latin squares has a convergent subsequence.
• Left-convergence and cut-distance convergence are equivalent for sequences of Latin squares, generalizing a key result from graph and permutation limits.
• A sampling lemma holds: a random Latin square of order $n$ drawn from a Latinon converges almost surely to the original Latinon as $n \to \infty$.
• Every Latinon can be approximated by a finite Latin square to arbitrary precision, a result established using Keevash’s existence theorems on combinatorial designs.
• The theory provides a complete framework for studying quasirandomness and extremal problems in Latin squares, as demonstrated by subsequent applications.
• The theory extends to higher-dimensional analogues, with potential for generalizing to higher-order permutations and hypergraphs.