The minimum size of critical sets in latin squares

Abstract

A critical set C of order n is a partial latin square of order n which is uniquely completable to a latin square, and omitting any entry of the partial latin square destroys this property. The size s( C) of a critical set C is the number of filled cells in the partial latin square. The size of a minimum critical set of order n is s( n). It is likely that s( n) is approximately 1 4 n 2 , though to date the best-known lower bound is that s( n) ⩾ n + 1. In this paper, we obtain some conditions on C which force s(C) ⩾ [ (n − 1) 2 ] 2 . For n > 20, this is used to show that in general s(n) ⩾ [ (7n − 3) 6 ] , thus improving the best-known result.