On avoiding odd partial Latin squares and r-multi Latin squares

We show that for any positive integer k 4, if R is a (2k − 1)× (2k − 1) partial Latin square, then R is avoidable given that R contains an empty row, thus extending a theorem of Chetwynd and Rhodes. We also present the idea of avoidability in the setting of partial r-multi Latin squares, and give some partial fillings which are avoidable. In particular, we show that ifR contains at most nr/2 symbols and if there is an n× n Latin squareL such that n of the symbols inL cover the filled cells inR where 0< < 1, then R is avoidable provided r is large enough. © 2006 Elsevier B.V. All rights reserved.