A set of positive integers D is called lonely if there exist real numbers α, δ ∈ (0, 1) such that each point of the dilation αD is at distance at least δ from the nearest integer. We prove that for every lonely set there is a 2-coloring of the integers without arbitrarily long monochromatic arithmetic progressions with steps d ∈ D. This result is a step towards a more general conjecture by Brown, Graham, and Landman, stating that a similar 2-coloring exists whenever the set of allowable steps D violates the restricted version of van der Waerden’s theorem.
展开▼
机译：如果存在真实的数字α，Δ∈（0,1），则将一组正整数D称为孤独，使得扩张αd的每个点处于从最近整数的距离至少Δ处。 我们证明，对于每一个孤独的设置，整数的整数有2个着色，没有任意长的单色算术进展与步骤d∈D。这一结果是朝着棕色，格雷厄姆和兰德曼更一般猜想的一步，说明了类似的 每当允许步骤D违反Van der Waerden的定理的限制版本时，存在2色。
展开▼
