A survey of finite embedding theorems for partial latin squares and quasigroups

Abstract

A latin square is an n × n array such that in each row and column each of the integers 1,2,3,...,n occurs exactly once. A partial latin square is an n × n array such that in each row and column each of the integers 1,2,...,n occurs at most once. An example of a 4 × 4 partial latin square is given below.

An immediate observation shows that the partial latin square P cannot be completed to a latin square; i.e., the empty cells cannot be filled in with numbers from the set {1,2,3,4} so that the result is a latin square. This is because as soon as cell (3,4) is filled in a contradiction arises. A very obvious question to ask at this point is whether or not it is possible to complete P if we are allowed to enlarge P and introduce additional symbols. That is, does there exist an m × m latin square Q agreeing with P in its upper left hand corner? (P is said to be embedded in Q). In a by now classic paper, Trevor Evans has shown that this is always possible. In fact, Evans proved that any n × n partial latin square could be embedded in some t × t latin square for every t ≥ 2n..., the best possible result of this kind. Evans' paper is now generic. It has become the starting point for a fascinating collection of problems in the study of latin squares: the so-called finite embedding problems. It is this collection of problems that will be surveyed in this set of notes. These notes are reasonably self contained so that certain parts may be a bit pedantic..., hopefully not too much so. Examples are included at every opportunity to (hopefully) illustrate what is going on.