On a Generalization of the van der Waerden Theorem

For a given length and a given degree and an arbitrary partition of the positive integers, there always is a cell containing a polynomial progression of that length and that degree; moreover, the coefficients of the generating polynomial can be chosen from a given semigroup and one can prescribe the occurring powers. A multidimensional version is included.Comment: 5 page

Superfilters, Ramsey theory, and van der Waerden's Theorem

Superfilters are generalized ultrafilters, which capture the underlying concept in Ramsey theoretic theorems such as van der Waerden's Theorem. We establish several properties of superfilters, which generalize both Ramsey's Theorem and its variant for ultrafilters on the natural numbers. We use them to confirm a conjecture of Ko\v{c}inac and Di Maio, which is a generalization of a Ramsey theoretic result of Scheepers, concerning selections from open covers. Following Bergelson and Hindman's 1989 Theorem, we present a new simultaneous generalization of the theorems of Ramsey, van der Waerden, Schur, Folkman-Rado-Sanders, Rado, and others, where the colored sets can be much smaller than the full set of natural numbers.Comment: Among other things, the results of this paper imply (using its one-dimensional version) a higher-dimensional version of the Green-Tao Theorem on arithmetic progressions in the primes. The bibliography is now update

New Lower Bounds for van der Waerden Numbers Using Distributed Computing

This paper provides new lower bounds for van der Waerden numbers. The number $W(k,r)$ is defined to be the smallest integer $n$ for which any $r$-coloring of the integers $0 \ldots, n-1$ admits monochromatic arithmetic progression of length $k$; its existence is implied by van der Waerden's Theorem. We exhibit $r$-colorings of $0\ldots n-1$ that do not contain monochromatic arithmetic progressions of length $k$ to prove that $W(k, r)>n$. These colorings are constructed using existing techniques. Rabung's method, given a prime $p$ and a primitive root $\rho$, applies a color given by the discrete logarithm base $\rho$ mod $r$ and concatenates $k-1$ copies. We also used Herwig et al's Cyclic Zipper Method, which doubles or quadruples the length of a coloring, with the faster check of Rabung and Lotts. We were able to check larger primes than previous results, employing around 2 teraflops of computing power for 12 months through distributed computing by over 500 volunteers. This allowed us to check all primes through 950 million, compared to 10 million by Rabung and Lotts. Our lower bounds appear to grow roughly exponentially in $k$. Given that these constructions produce tight lower bounds for known van der Waerden numbers, this data suggests that exact van der Waerden Numbers grow exponentially in $k$ with ratio $r$ asymptotically, which is a new conjecture, according to Graham.Comment: 8 pages, 1 figure. This version reflects new results and reader comment

On the optimality of the uniform random strategy

The concept of biased Maker-Breaker games, introduced by Chv\'atal and Erd{\H o}s, is a central topic in the field of positional games, with deep connections to the theory of random structures. For any given hypergraph ${\cal H}$ the main questions is to determine the smallest bias $q({\cal H})$ that allows Breaker to force that Maker ends up with an independent set of ${\cal H}$. Here we prove matching general winning criteria for Maker and Breaker when the game hypergraph satisfies a couple of natural `container-type' regularity conditions about the degree of subsets of its vertices. This will enable us to derive a hypergraph generalization of the $H$-building games, studied for graphs by Bednarska and {\L}uczak. Furthermore, we investigate the biased version of generalizations of the van der Waerden games introduced by Beck. We refer to these generalizations as Rado games and determine their threshold bias up to constant factors by applying our general criteria. We find it quite remarkable that a purely game theoretic deterministic approach provides the right order of magnitude for such a wide variety of hypergraphs, when the generalizations to hypergraphs in the analogous setup of sparse random discrete structures are usually quite challenging.Comment: 26 page