A Five Color Zero-Sum Generalization

Let g_(zs) (m, 2k) (g_(zs) (m, 2k+1)) be the minimal integer such that for any coloring Δ of the integers from 1, . . . , g_(zs) (m, 2k) by ∪+^k_(i=1)Z^i_m (the integers from 1 to g_(zs) (m, 2k+1) by ∪+^k_(i=1) Z^i_m ∪ {∞ }) there exist integers x₁ < … < x_m < y₁ < … y_m such that 1. there exists j_x such that Δ(x_i) ∈ Z^(jx)_m for each i and ∑_(i =1)^m Δ(x_i) = 0 mod m (or Δ(x_i) = ∞ for each i); 2. there exists j_y such that Δ(y i) ∈ Z^(jy)_m for each i and ∑_(i =1)^m Δ(y_i) = 0 mod m (or Δ(y_i) = ∞ for each i); and 1. 2(x_m −x₁) ≤ y_m −x₁. In this note we show g_(zs) (m, 2) = 5m−4 for m ≥ 2, g_(zs) (m, 3) = 7m+[m/2]−6 for m ≥ 4, g_(zs) (m, 4) = 10m−9 for m ≥ 3, and g_(zs) (m, 5) = 13m−2 for m ≥ 2.