# Dinitz Conjecture extension to rectangles

The Dinitz Conjecture, which was proved later in a more general form by Galvin, stated that given an $$n\times n$$ array, its elements could be filled exactly like a latin square, where the elements in each row/column is chosen from a set (list) of $$m\ge n$$ symbols.

Whether an extension of this is possible for partial latin rectangles? That is given a rectangle of size $$m\times n,\ \ m\neq n$$ and symbols $$\ge max(m,n)$$, is it possible to fill $$min(m,n)$$ symbols in each row and/or column such that the symbols in each row, column as well as diagonal (the places $$a_{ij}$$ with $$j=i$$, where $$i,j$$ denote row and column indices and $$a$$ denotes the symbol) are distinct?

• @RebeccaJ.Stones yes, the latin rectangle is partial, that is, has some empty cells Sep 13 '19 at 11:55
• I don't understand the statement. What does your conjecture say in the special case where $m=n$? It's not the Dinitz Conjecture because the DC does not require distinct symbold on the diagonal, and I believe it would be false if you just added that requirement. Also not sure what "fill a maximum of $\min(m,n)$ symbols . . ." means. If it means "fill at most $\min(m,n)$ symbols," can't you satisfy it trivially by leaving all cells empty?
– bof
Oct 7 '19 at 3:42
• @bof thanks for that observation. Edited the question now. Oct 7 '19 at 19:28