Small Partial Latin Squares that Cannot be Embedded in a Cayley Table

Abstract

We answer a question posed by D\'enes and Keedwell that is equivalent to the following. For each order $n$ what is the smallest size of a partial latin square that cannot be embedded into the Cayley table of any group of order $n$? We also solve some variants of this question and in each case classify the smallest examples that cannot be embedded. We close with a question about embedding of diagonal partial latin squares in Cayley tables.