A Multiplicity Problem Related to Schur Numbers

For each natural number n, let Cn represent the set of all 2-colorings of the set {1, 2, . . . , n}. Given a natural number n and a coloring ∆ ∈ Cn, let S(∆) represent the set S(∆) = {x3 | ∃ x1, x2 s.t. x1 + x2 = x3 and ∆(x1) = ∆(x2) = ∆(x3)}. Given a natural number n, let f(n) = min ∆∈Cn S(∆). For all natural numbers n and r where n2 ≤ r ≤ n, let Cn,r represent the set of all 2-colorings of the set {1, 2, . . . , n} where max{|∆−1(0)|, |∆−1(1)|} = r. Given natural numbers n and r where n2 ≤ r ≤ n, let f(n, r) = min ∆∈Cn,r S(∆). In this paper it is determined that for all natural numbers n, f(n) = { 0 1 ≤ n ≤ 4 ⌊ n−3 2 ⌋ n ≥ 5 and for all natural numbers n and r where n ≥ 5 and n2 ≤ r ≤ n, f(n, r) = { r − 2 r < n r − 1 r = n.