In an array of dots, how many can be colored red without forming a red rectangle? • We use rectangular arrays of dots, and consider only those rectangles with horizontal and vertical sides. A dot-rectangle is called “red” if its four corners are red. (Other dots can be of any color.) For whole numbers $m, n,$ define $R(m, n)$ to be the largest number of dots in an m ×n array that can be colored red without making a red rectangle. For instance, in a $3\times 3$ array an L-shaped arrangement of 5 red dots contains no red rectangle. Therefore $R(3,3)\ge 5$ But 5 is not maximal: there is a rectangle-free arrangement of 6 red dots, proving that $(3,3)\ge 6$ After some work I managed to prove: Every arrangement of 7 red dots in a $3\times 3$ array must include a red rectangle. This shows that $R(3,3)=6$
Question: Investigate other values of $R(m, n)$, observing patterns, making conjectures, and constructing proofs