Partial Latin Square

In a normal $n\times n$ Latin Square, the entries in each row and column are chosen from a ``global'' set of $n$ objects. Like a Latin square, a partial Latin square has no two rows or columns which contain the same two symbols. However, in a partial Latin square, each cell is assigned one of its own set of $n$ possible ``local'' (and distinct) symbols, chosen from an overall set of more than three distinct symbols, and these symbols may vary from location to location. For example, given the possible symbols $\{$1, 2, ..., 6$\}$ which must be arranged as

$$\matrix{ \{1, 2, 3\} & \{1, 3, 4\} & \{2, 5, 6\}\cr \{2, 3, ... ... & \{4, 5, 6\}\cr \{4, 3, 6\} & \{3, 5, 6\} & \{2, 3, 5\},\cr}$$

the $3\times 3$ partial Latin square

$$\matrix{1 & 3 & 2\cr 2 & 4 & 5\cr 6 & 5 & 3\cr}$$

can be constructed.

See also Dinitz Problem, Latin Square

References

Cipra, B. ``Quite Easily Done.'' In What's Happening in the Mathematical Sciences 2, pp. 41-46, 1994.