Zero-Sum Ascending Waves

A sequence of positive integers \(a_1 \leq a_2 \leq \ldots \leq a_n\) is called an ascending monotone wave of length \(n\), if \(a_{i+1} – a_{i} \geq a_{i} – a_{i-1}\) for \(i = 2, \ldots, n-1\). If \(a_{i+1} – a_{i} > a_{i} – a_{i-1}\) for all \(i = 2, \ldots, n-1\) the sequence is called an ascending strong monotone wave of length \(n\). Let \({Z}_k\) denote the cyclic group of order \(k\). If \(k | n\), then we define \(MW(n, {Z}_k)\) as the least integer \(m\) such that for any coloring \(f : \{1, \ldots, m\} \to {Z}_k\), there exists an ascending monotone wave of length \(n\), where \(a_n \leq m\), such that \(\sum_{i=1}^n f(a_i) = 0 \mod k\). Similarly, define \(SMW(n, {Z}_k)\), where the ascending monotone wave in \(MW(n, {Z}_k)\) is replaced by an ascending strong monotone wave. The main results of this paper are:

1. \(\frac{\sqrt{k}}{2}n \leq MW(n, Z_k) \leq c_1(k)n\). Hence, this result is tight up to a constant factor which depends only on \(k\).
2. \(\binom{n}{2} < SMW(n, {Z}_k) \leq c_2(k)n^2\). Hence, this result is tight up to a constant factor which depends only on \(k\).
3. \(MW(n, {Z}_2) = {3n}/{2}\).
4. \(\frac{23}{12}n – {7}/{6} \leq MW(n, {Z}_3) \leq 2n+3\).

These results are the zero-sum analogs of theorems proved in [1] and [5].