The Gap Structure of a Family of Integer Subsets

  • AndrĂ© Bernardino
  • Rui Pacheco
  • Manuel Silva
Keywords: piecewise syndetic, complementing pairs, Brown's lemma, Ramsey theory


In this paper we investigate the gap structure of a certain family of subsets of $\mathbb{N}$ which produces counterexamples both to the "density version" and the "canonical version" of Brown's lemma. This family includes the members of all complementing pairs of $\mathbb{N}$. We will also relate the asymptotical gap structure of subsets of $\mathbb{N}$ with their density and investigate the asymptotical gap structure of monochromatic and rainbow sets with respect to arbitrary infinite colorings of $\mathbb{N}$.
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