ON THE CYCLIC VAN DER WAERDEN NUMBERS

摘要

Suppose k, r ∈ Z and k ≥ 3 and r > 2. The Van der Waer-den number W (k; r) is the smallest positive integer N such that every coloring of {0, ... , N — 1} with r or fewer colors contains a monochromatic k-term arithmetic progression. That is, there exists a monochromatic subset {a, a + d, ... , a + (k — 1)d}∩{0, ... , N — 1}, where d > 0. These numbers are well defined by a famous theorem of B. van der Waerden; see [3] for a full account. Very few of these numbers are known exactly. For the purposes of this note, it will suffice to know that W(3; 2) = 9, as detailed in [2].