Monochromatic Quasi-Progressions and Monochromatic Sequences Whose Gaps Belong to {d, 2d, ... , md}
by
Bruce Landman
University of North Carolina at Greensboro

A quasi-progression of diameter n is a sequence {x₁, ... , x_k} for which there exists a positive integer L such that L <= x_i-x_i-1 <= L+n for i=2, ... , k. Define an h_m-progression to be a sequence {x₁, ... , x_k} such that for some d in Z⁺, x_i+1-x_i in {d, 2d, ... , md} for i=2, ... , k. Let Q_n(k) and h_m(k) represent the associated van der Waerden-like numbers for k-term quasi-progessions of diameter n and for k-term h_m-progressions, respectively. Upper and lower bounds for Q_n(k) and h_m(k) are found for certain cases.

Date received: April 27, 1998