Let r and k be positive integers with r | k. Denote by S_3 (k; r) the minimum integer n such that every coloring χ : [1, n] → {0,1,..r — 1} admits a solution to ∑_(i=1)~(k-1)x_i =x_k with _(i=1)χ(x_i) ≡ 0 (modr). We give some formulas and lower bounds for various instances.
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