3. Recall that R(2;3, 4) = 9 means that any edge 2-colouring of K9 yields a K3 monochromatic in the first colour or a K4 monochromatic in the second colour. Let L3 denote the equation x1 + X2 = X3 and let L4 denote the equation x] + x2 + X3 = X4. (a) Adjust the proof of Schur's Theorem demonstrated in the class to prove that any 2-colouring c : [1,9] -> { } will yield a .- coloured solution of [3 or a -coloured solution of L4. (b) Would any 2-colouring of [1, 8] have the same property? What about [1, 7]? [1, 6]? 4. (a) Prove that for any given finite colouring of positive integers, there is an infinite number of monochromatic Schur triples. (b) Is it true that for any given finite colouring of positive integers, any partition regular equation has an infinite number of monochromatic solutions?

Fig: 1