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 if R contains at most nr/2 symbols and if there is an n×n Latin square L such that δn of the symbols in L cover the filled cells in R where 0<δ<1, then R is avoidable provided r is large enough.