... to the left of q, and q is to the right of p. In a similar vein,if p and q are vertices of the game board such that p < q, then we define the interval[p, q] to be the intersection of the ... arbitrary constant.Proof: Our proof proceeds exactly like the proof of Theorem 1, except instead of using the Erd˝os–Selfridge theorem on each sub-board of size (mk)d, we use a modified form of Theorem ... sub-board to the next is due to a progression in at leastone of the dimensions. There are only d dimensions, so there can be at most d switches.If there were d+1 switches, then the line would...