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

Abstract

We show that for any positive integer k ⩾ 4 , if R is a ( 2 k - 1 ) × ( 2 k - 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 δ n of the symbols in L cover the filled cells in R where 0 < δ < 1 , then R is avoidable provided r is large enough.