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On Asymmetric Colorings of Integer Grids

Let ν(Z) be the minimal number of colors enough to color the m-dimensional integer grid Z so that there would be no infinite monochromatic symmetric subsets. Banakh and Protasov [3] compute ν(Z) = m + 1. For the one-dimensional case this just means that one can color positive integers in red, while negative in blue, thereby avoiding an infinite monochromatic symmetric subset by trivial reason. This motivates the question what changes if we allow only colorings unlimited in both directions (in “all” directions for m > 1). In this paper we show that then ν(Z) increases in 1, whereas for higher dimensions the values ν(Z) remain unaffected. Furthermore we examine the density properties of a set A ⊆ Z that ensure the existence of infinite symmetric subsets or arbitrarily large finite symmetric subsets in A. In the case that A is a sequence with small gaps, we prove a multi-dimensional analogue of the Szemeredi theorem, with symmetric subsets in place of arithmetic progressions. A similar two-dimensional statement is known for collinear subsets (Pomerance [10]), whereas for two-dimensional arithmetic progressions even the corresponding version of van der Waerden’s theorem is known to be false. We also observe that A ⊆ N contains arbitrarily large symmetric subsets whenever the series ∑ a∈A 1/a diverges (for arithmetic progressions this is the well-known unproven conjecture of Erdős). A natural two-dimensional analogue of the latter statement is false. We show a counterexample built upon Erdős’s construction of a dense infinite B2-sequence. ∗Department of Mechanics & Mathematics, Lviv University, Universytetska 1, 290602 Lviv, Ukraine. E-mail: tbanakh@franko.lviv.ua. Research supported in part by grant INTAS-96-0753. †Department of Numerical Mathematics & Programming, State University “Lvivska Polytechnika”, Bandera St. 12, 290646 Lviv, Ukraine. Part of this work was done while visiting the Institute of Mathematics, University of Vienna, supported by an OAD grant. ‡Part of this work was done while visiting the Institute of Information Systems, Vienna University of Technology, supported by a Lise Meitner Fellowship of the Austrian Science Foundation (FWF). Current e-mail: oleg@dbai.tuwien.ac.at.