Constrained completion of partial latin squares

In this paper, we combine the notions of completing and avoiding partial latin squares. Let PP be a partial latin square of order nn and let QQ be the set of partial latin squares of order nn that avoid PP. We say that PP is QQ-completable if PP can be completed to a latin square that avoids Q∈QQ∈Q. We prove that if PP has order 4t4t and contains at most t−1t−1 entries, then PP is QQ-completable for each Q∈QQ∈Q when t≥9t≥9.