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. c. Next we prove the following claim by induction on k: For all r, k E N there exists a natural number n(r, k) such that for any r-colouring x of [1, n(r, k)] there exist aj, a2, ..., Ak such that for any nonempty subset I of the set [1, k] a(I) = Xa; [1, n] and x(a(1)) = x(Qmax (7)). iel o Prove the base case, i.e prove that that n(r, 1) exists. o For the inductive step suppose that k > 1 is such that there exists a natural number n(r, k) such that for any r-colouring x of [1, n) there exist a1, 22, ..., ak such that for any nonempty subset I of the set [1, k] a(I) = Xa; [1, n] and x(a(1)) = x(Qmax (n)). iel Let N = 2.W(r, n(r, k) + 1), where W(r, n(r, k) + 1) is the van der Waerden number that guarantees the existence of a monochromatic (n(r, k) + 1)-term arithmetic progression in every r-coloured interval that contains W(r, n(r, k) + 1) consecutive positive integers. Fix an r-colouring & of the interval [1, N]. Part 1: Prove that there are akt1 [ + 1, N) and d e N such that the arithmetic progression {ak+1 +j.d:05 ; 5 n(r, k)}c [A + 1, N) is &- monochromatic. + Part 2: Let d be as above. Explain why there exist al, 22, ..., ak E {d, 2d, ...,n(r, k).d} such that for any nonempty subset I of the set [1, k] a(I) = Xa; {d, 2d, ... , n(r,k) d} and &(a(1) = $(amax ()). iel Time left Part 3: Complete the proof of the inductive step. 3 days, 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. c. Next we prove the following claim by induction on k: For all r, k E N there exists a natural number n(r, k) such that for any r-colouring x of [1, n(r, k)] there exist aj, a2, ..., Ak such that for any nonempty subset I of the set [1, k] a(I) = Xa; [1, n] and x(a(1)) = x(Qmax (7)). iel o Prove the base case, i.e prove that that n(r, 1) exists. o For the inductive step suppose that k > 1 is such that there exists a natural number n(r, k) such that for any r-colouring x of [1, n) there exist a1, 22, ..., ak such that for any nonempty subset I of the set [1, k] a(I) = Xa; [1, n] and x(a(1)) = x(Qmax (n)). iel Let N = 2.W(r, n(r, k) + 1), where W(r, n(r, k) + 1) is the van der Waerden number that guarantees the existence of a monochromatic (n(r, k) + 1)-term arithmetic progression in every r-coloured interval that contains W(r, n(r, k) + 1) consecutive positive integers. Fix an r-colouring & of the interval [1, N]. Part 1: Prove that there are akt1 [ + 1, N) and d e N such that the arithmetic progression {ak+1 +j.d:05 ; 5 n(r, k)}c [A + 1, N) is &- monochromatic. + Part 2: Let d be as above. Explain why there exist al, 22, ..., ak E {d, 2d, ...,n(r, k).d} such that for any nonempty subset I of the set [1, k] a(I) = Xa; {d, 2d, ... , n(r,k) d} and &(a(1) = $(amax ()). iel Time left Part 3: Complete the proof of the inductive step. 3 days