A blurred view of Van der Waerden type theorems

Purely Combinatorial Proofs of Van Der Waerden-Type Theorems

A van der Waerden Variant

The classical van der Waerden Theorem says that for every every finite set S of natural numbers and every k-coloring of the natural numbers, there is a monochromatic set of the form aS+b for some a > 0 and b ≥ 0. I.e., monochromatism is obtained by a dilation followed by a translation. We investigate the effect of reversing the order of dilation and translation. S has the variant van der Waerde...

Sane Bounds on Van der Waerden-Type Numbers

In Ramsey theory (a branch of combinatorics) one often proves that a function exists by giving enormous bounds on it. The function’s actual values may be much smaller; however, this is not known. In this paper we explore variants and special cases of these functions where we obtain much smaller bounds. The following is known: no matter how the lattice points of a coordinate plane are colored re...

Bounds on some van der Waerden numbers

For positive integers s and k1,k2, . . . ,ks, the van der Waerden number w(k1,k2, . . . ,ks;s) is the minimum integer n such that for every s-coloring of set {1,2, . . . ,n}, with colors 1,2, . . . ,s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w(k,m;2) for fixed m. We include a table of values of w(k,3;2) that are very close to this l...

Satisfiability and Computing van der Waerden Numbers

In this paper we bring together the areas of combinatorics and propositional satisfiability. Many combinatorial theorems establish, often constructively, the existence of positive integer functions, without actually providing their closed algebraic form or tight lower and upper bounds. The area of Ramsey theory is especially rich in such results. Using the problem of computing van der Waerden n...