Progressions in sequences of nearly consecutive integers (Q1268731)

For \(k\geq 3\) and \(r\geq 2\), let \(G(r,k)\) denote the smallest positive integer \(m\) such that every sequence of integers \(a_1, \cdots , a_m\) such that \(1\leq a_{j+1}-a_j\leq r\), \(1\leq j\leq m-1\), contains a \(k\)-term arithmetic progression. The authors show that for every fixed \(r\geq 2\), there is a constant \(c_r\), such that \(G(k,r)>r^{k-c_r\sqrt{k}}\), for all \(k\).