AI 中文总结
研究奥因斯和集问题，通过证明对自然数二染色存在无限子集\(B\)使\(B + B\)单色，给出肯定回答，并呈现了相关加权形式。
AI 中文摘要
本文对奥因斯和集问题给出肯定回答。即证明对于自然数的任何二染色，存在无限子集\(B\subset \mathbb{N}\)使得\(B + B\)是单色的。同时还给出了该结果的一些加权形式。
英文摘要
We give an affirmative answer to Owings's sumset question: for any $2$-coloring of natural numbers, there is an infinite $B\subseteq\mathbb{N}$ such that $B+B$ is monochromatic. More generally, for every $m,\ell\in\mathbb{N}$ and every $2$-coloring of $\mathbb{N}$, there is an infinite $B\subseteq\mathbb{N}$ such that $$ (m+\ell)B\cup\{mx+\ell y:x,y\in B,\ x<y\} $$ is monochromatic.
CommentsIn the newest version, we resolve the weighted form of Owings's sumset question completely
