Infinite unrestricted sumsets of the form B+B$B+B$ in sets with large density

Abstract For a set , we characterize the existence of an infinite set and such that , where , in terms of the density of the set . Specifically, when the lower density or the upper density , the existence of such a set and is assured. Furthermore, whenever or , we show that the shift is unnecessary and we also provide examples to show that these bounds are sharp. Finally, we construct a syndetic three‐coloring of the natural numbers that does not contain a monochromatic for any infinite set and number .