Coloring of a table with red or green rows and columns

Solution 1:

Suppose there are at least 11 good rows and columns. there must be at least 1 good row and at least 1 good column. this row and column must have the same color because they intersect. WLOG say they are red. the red row precludes green columns and the red column precludes green rows, so there cannot be any green rows or columns. suppose there are $m$ red rows and $n$ red columns. $$10m+10n-mn\leq 50$$ $$(10-m)(10-n)\geq 50$$ $$\frac{(10-m)+(10-n)}{2}\geq \sqrt{(10-m)(10-n)}\geq 5\sqrt2$$ $$m+n\leq 20-10\sqrt2<6$$ contradiction. so there are at most 10 good rows and columns. the construction is 5 green, 5 red, all parallel to one another.