say a color taints a row or a column if at least three cells of that row or column are of that color. The negation of the problem statement implies all rows and columns are tainted, but note each color can only taint at most one line and one column. Since we have ten lines and columns, this implies that each color taints exactly one row and one column. Indeed we can paint the cells such that each color taints exactly one row and one column, but in this case each row and column is forced to have exactly three colors used.

The general version can be solved similarly