A superlinear lower bound for the size of a critical set in a latin square

Abstract

AbstractA critical set is a partial latin square that has a unique completion to a latin square, and is minimal with respect to this property. Let scs(n) denote the smallest possible size of a critical set in a latin square of order n. We show that for all n, $scs(n)\geq n\lfloor (\log{n})^{1/3}/2\rfloor$. Thus scs(n) is superlinear with respect to n. We also show that scs(n) ≥ 2n−32 and if n ≥ 25, ${\rm scs}(n)\geq \lceil (3n-7)/2 \rceil$. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 269–282, 2007