Lower Bounds on van der Waerden Numbers: Randomized- and Deterministic-Constructive

The van der Waerden number W (k, 2) is the smallest integer n such that every 2-coloring of 1 to n has a monochromatic arithmetic progression of length k. The existence of such an n for any k is due to van der Waerden but known upper bounds on W (k, 2) are enormous. Much effort was put into developing lower bounds on W (k, 2). Most of these lower bound proofs employ the probabilistic method often in combination with the Lovász Local Lemma. While these proofs show the existence of a 2-coloring that has no monochromatic arithmetic progression of length k they provide no efficient algorithm to find such a coloring. These kind of proofs are often informally called nonconstructive in contrast to constructive proofs that provide an efficient algorithm. This paper clarifies these notions and gives definitions for deterministicand randomized-constructive proofs as different types of constructive proofs. We then survey the literature on lower bounds on W (k, 2) in this light. We show how known nonconstructive lower bound proofs based on the Lovász Local Lemma can be made randomized-constructive using the recent algorithms of Moser and Tardos. We also use a derandomization of Chandrasekaran, Goyal and Haeupler to transform these proofs into deterministic-constructive proofs. We provide greatly simplified and fully self-contained proofs and descriptions for these algorithms.