On the history of van der Waerden’s theorem on arithmetic progressions

In this expository note, we discuss the celebrated theorem known as “van der Waerden’s theorem on arithemetic progressions,” the history of work on upper and lower bounds for the function associated with this theorem, a number of generalizations, and some open problems. 1 van der Waerden’s theorem, and the function w(k) The famous theorem of van der Waerden on arithmetic progressions is usually stated in the following way. Theorem 1. (van der Waerden 1927 [24]). For every positive integer k, there exists a positive integer n such that if the set [1;n = f1;2; : : : ;ng] is partitioned into two subsets, then at least one of the subsets must contain an arithmetic progression of size k. (Recall that an arithmetic progression of size k is a set of the form fa;a+d;a+2d; : : : ;a + (k 1)dg, where d > 0.) This latter statement (with r subsets instead of two subsets) is the statement given in van der Waerden’s original proof, which used a double induction on k and r. Van der Waerden’s original proof was found with the help of Artin and Schrier. See [25] for a nice description of how the proof was found, and of the proof itself. Good expositions of this proof are given in the charming book by Khinchin [14], and in the book by Graham, Rothschild, and Spencer [13]. Other proofs of this statement can be found in [1, 7, 15, 22]. Perhaps the easiest of these to read is [15]. A very short proof is in [13]. A topological proof is in [9]. An algebraic proof is in [2]. For each positive integer k, we let w(k) denote the smallest positive integer such that if the set [1;w(k)] is partitioned into two subsets, then at least one of the subsets must contain an arithmetic progression of size k. The function w(k) is often called the van der Waerden function. Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada V3G 1G4. [email protected] †Department of Mathematical Sciences, University of Nevada, Las Vegas, NV, USA 89154-4020. [email protected]