Arithmetic Progressions in Sequences with Bounded Gaps

LetG(k, r) denote the smallest positive integergsuch that if 1=a1, a2, …, agis a strictly increasing sequence of integers with bounded gapsaj+1−aj⩽r, 1⩽j⩽g−1, then {a1, a2, …, ag} contains ak-term arithmetic progression. It is shown thatG(k, 2)>(k−1)/2 (43)(k−1)/2,G(k, 3)>(2k−2/ek)(1+o(1)),G(k, 2r−1)>(rk−2/ek)(1+o(1)),r⩾2.