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

We show that for any positive integer k⩾4k⩾4, if RR is a (2k-1)×(2k-1)(2k-1)×(2k-1) partial Latin square, then RR is avoidable given that RR 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 RR contains at most nr/2nr/2 symbols and if there is an n×nn×n Latin square LL such that δnδn of the symbols in LL cover the filled cells in RR where 0<δ<10<δ<1, then RR is avoidable provided r is large enough.