Zero-Sum Ascending Waves

A sequence of positive integers a1 ≤ a2 ≤ . . . ≤ an is called an ascending monotone wave of length n, if ai+1−ai ≥ ai−ai−1 for i = 2, . . . , n−1. If ai+1−ai > ai−ai−1 for all i = 2, . . . , n−1 the sequence is called an ascending strong monotone wave of length n. Let Zk denote the cyclic group of order k. If k | n, then we define MW (n,Zk) as the least integer m such that for any coloring f : {1, . . . ,m} → Zk there exists an ascending monotone wave of length n, where an ≤ m, such that ∑n i=1 f(ai) ≡ 0 mod k. Similarly, define SMW (n,Zk), where the ascending monotone wave in MW (n,Zk) is replaced by an ascending strong monotone wave. The main results of this paper are: • √ k 2 n ≤ MW (n,Zk) ≤ c1(k)n. Hence, this result is tight up to a constant factor which depends only on k. • ( n 2 ) < SMW (n,Zk) ≤ c2(k)n. Hence, this result is tight up to a constant factor which depends only on k. • MW (n,Z2) = 3n/2. • 23 12n− 7/6 ≤MW (n,Z3) ≤ 2n + 3. These results are the zero-sum analogs of theorems proved in [1] and [5]. AMS 1991 Mathematics subject classification: 05D10.