The gap structure of a family of integer subsets (Q405142)

Summary: 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 \(\mathbb{N}\).