Progressions in sequences of nearly consecutive integers

For k > 2 and r greater than or equal to 2, let G(k, r) denote the smallest positive integer g such that every increasing sequence of g integers {a(1), a(2), ..., a(g)} with gaps a(j + 1) - a(j) is an element of{1, ..., r}, 1 less than or equal to j less than or equal to g - 1 contains a k-term arithmetic progression. Brown and Hare proved that G(k, 2) > root(k - 1)/2 (4/3)((k - 1)/2) and that G(k, 2s - 1) > (s(k - 2)/ek)(1 + o(1)) for all s greater than or equal to 2. Here we improve these bounds and prove that G(k, 2) > 2(k - O(root k)) and, more generally, that for every fixed r greater than or equal to 2 there exists a constant c(r) > 0 such that G(k, r) > r(k - cr root k) for all k. (C) 1998 Academic Press