Upper and lower bounds for some rado numbers for inequalities

Abstract

Let an integer m ≥ 2 be given and let positive integers c₁, C₂, . . . , C_m-1 be given. Let n = f(m: c₁, C₂, . . . , c_m-1) be the least integer such that for every coloring, Δ: {1, 2, . . . , n} → {0, 1}, there exist m integers, cursive Greek chi₁, cursive Greek chi₂, . . . , cursive Greek chi_m, such that C₁cursive Greek chi₁ + C₂cursive Greek chi₂ + ⋯ + C_m-1cursive Greek chi_m-1 < cursive Greek chi_m, cursive Greek chi₁ < cursive Greek chi₂ < ⋯ < cursive Greek chi_m and Δ(cursive Greek chi₁) = Δ(cursive Greek chi₂) = ⋯ = Δ(cursive Greek chi_m,). In this paper, upper and lower bounds for f(m: C₁, C₂, . . . , C_m-1) are found and the exact value is given for some specific cases.