Nodes of the infinite checkered paper are painted in two colors. Prove that there are two horizontal and two vertical lines which intersect in the points of one color.
Take three vertical and nine horizontal lines. We will consider only the points of intersection of these lines. Since there are only 23=8 different colorings of the three points into two colors, there are two horizontal lines on which there are identically painted triplets of points. Among the three points painted into two colors there are two equally colored points. Vertical lines through these points, together with the previously selected two horizontal lines are the sought-for.