Teaching dimension, VC dimension, and critical sets in Latin squares

TL;DR

This paper establishes new lower bounds on the size of critical sets in Latin squares, confirming their quadratic order, and explores related learning-theoretic dimensions, providing insights into their complexity.

Contribution

It proves a quadratic lower bound for the smallest critical set size in Latin squares and analyzes related learning-theoretic dimensions, improving previous bounds.

Findings

Lower bound of n^2/10^4 for critical set size in Latin squares

Confirmation of quadratic order of minimal critical sets

Lower bounds on VC-dimension and recursive teaching dimension of Latin squares

Abstract

A critical set in an $n \times n$ Latin square is a minimal set of entries that uniquely identifies it among all Latin squares of the same size. It is conjectured by Nelder in 1979, and later independently by Mahmoodian, and Bate and van Rees that the size of the smallest critical set is $\lfloor n^2/4\rfloor$. We prove a lower-bound of $n^2/10^4$ for sufficiently large $n$, and thus confirm the quadratic order predicted by the conjecture. We prove a lower-bound of $n^2/10^4$ for sufficiently large $n$, and thus confirm the quadratic order predicted by the conjecture. This improves a recent lower-bound of $\Omega(n^{3/2})$ due to Cavenagh and Ramadurai. From the point of view of computational learning theory, the size of the smallest critical set corresponds to the minimum teaching dimension of the set of Latin squares. We study two related notions of dimension from learning theory. We prove a lower-bound of $n^2-(e+o(1))n^{5/3}$ for both of the VC-dimension and the recursive teaching dimension.