A note on monoton waves (Q1119693)

A sequence \(x_ 0<...<x_{k-1}\) of nonnegative integers is called (i) an ascending wave if \(x_ i-x_{i-1}\leq x_{i+1}-x_ i\) for \(0<i<k- 1\), and (ii) a descending wave if \(x_ i-x_{i-1}\geq x_{i+1}-x_ i\) for \(0<i<k-1\). For positive integers \(k,\ell \geq 2\) let \(\omega(k,\ell)\) denote the least positive integer such that for every integer \(n\geq \omega (k,\ell)\) every integral sequence \(0\leq x_ 0<...<x_{n-1}\) contains a subsequence which is either a k-term ascending wave or an \(\ell\)-term descending wave. Using the weak version of Ramsey's theorem the author proves that \[ \omega (k,\ell)=\binom{k+\ell-4}{k-2}+\ell. \]