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

TL;DR

This paper reviews and clarifies the concepts of nonconstructive, randomized-constructive, and deterministic-constructive proofs for lower bounds on van der Waerden numbers, and demonstrates how to make existing probabilistic proofs constructive using recent algorithms.

Contribution

It introduces clear definitions for different types of constructive proofs and shows how to convert nonconstructive bounds into randomized- and deterministic-constructive bounds using recent algorithmic techniques.

Findings

Nonconstructive proofs can be made randomized-constructive with Moser-Tardos algorithms.

Derandomization techniques can transform these proofs into deterministic-constructive proofs.

Simplified, self-contained proofs of these algorithms are provided.

Abstract

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 Lov\'asz 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 deterministic- and 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 Lov\'asz 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.