The purpose of this exercise is to establish a proof of Folkman's theorem: For all r, k E N there exists a natural number M(r, k) such that for every r-colouring of [1, M] there exist 21, 22, ... , QE E [1, M] with all a; distinct, such that the set m F*(Q1, ... , am) = { 6;; : ; {0,1} and & + ... +&+0} i=0 is monochromatic. = d. To complete the proof of Folkman's theorem show that one can take M M(r, k) = n(r,r (k 1) + 1), where n(r, r (k 1) + 1) is the number guaranteed by the lemma proved in (c). Fix an r-colouring x of [1, M]. i. Justify the following claim: There exist al, 22, ..., , Ar(k-1)+1 such that for any nonempty subset I of the set [1,r(k 1) + 1] a(I) = a; [1, M] and x (a(I)) = x(Qmax (1)). iel ii. Define the r-colouring n of the set [1, r(k 1) + 1] in the following way: For any je [1, r(k 1) + 1] n(j) = x(a(1)) = x ( where 0 I C[1, r(k 1) + 1] and max(I) = j. iel Is the colouring n well defined? Why yes, or why not? iii. Prove that there is an n-monochromatic set S C [1, r(k 1) + 1] such that|S| = k. iv. Finish the proof of Folkman's theorem by proving that the set of all sums of the elements of the set A = {a; : i e C} is x-monochromatic. The purpose of this exercise is to establish a proof of Folkman's theorem: For all r, k E N there exists a natural number M(r, k) such that for every r-colouring of [1, M] there exist 21, 22, ... , QE E [1, M] with all a; distinct, such that the set m F*(Q1, ... , am) = { 6;; : ; {0,1} and & + ... +&+0} i=0 is monochromatic. = d. To complete the proof of Folkman's theorem show that one can take M M(r, k) = n(r,r (k 1) + 1), where n(r, r (k 1) + 1) is the number guaranteed by the lemma proved in (c). Fix an r-colouring x of [1, M]. i. Justify the following claim: There exist al, 22, ..., , Ar(k-1)+1 such that for any nonempty subset I of the set [1,r(k 1) + 1] a(I) = a; [1, M] and x (a(I)) = x(Qmax (1)). iel ii. Define the r-colouring n of the set [1, r(k 1) + 1] in the following way: For any je [1, r(k 1) + 1] n(j) = x(a(1)) = x ( where 0 I C[1, r(k 1) + 1] and max(I) = j. iel Is the colouring n well defined? Why yes, or why not? iii. Prove that there is an n-monochromatic set S C [1, r(k 1) + 1] such that|S| = k. iv. Finish the proof of Folkman's theorem by proving that the set of all sums of the elements of the set A = {a; : i e C} is x-monochromatic