Can you 4-color a 17x17 grid without getting four points the same color that form a rectangle?

We will be looking at colorings of grids. A c-coloring of a grid is an assignment toe every grid point a color. For example, this is a 2-coloring of the 3 x 7 grid using colors R,B,G.rrbbbrrbbrbrbbrrrrbrbNote that there is a rectangle with all four corners the same color (we use capital letters to denote it)RrbbbRrbbrbrbbRrrrbRbIf a grid can be c-colored without a monochromatic rectangle we say that the grid is c-colorable.Which grids can be 2-colored? 3-colored? 4-colored? etc.1) We have characterized EXACTLY which grids are 2-colorable.2) We have characterized EXACTLY which grids are 3-colorable.3) We have made porgress on EXACTLY which grids are 4-colorable.4) We have GIVEN UP on trying to find EXACTLY which grids are 5-colorable.The work is a combination of some clever math and some computer work. The questions has its origins in Ramsey Theory which we will also discuss.(William Gasarch will be teaching a grad course on Ramsey Theory in the Spring. This talk is an example of what will be covered.)