单色和与商近零
Monochromatic Sums and Quotients Near Zero
AI总结：
研究单色和与商近零的性质，证明了特定集合在密集子半群下为单色，且展示了特定模式的分区正则性。
摘要：
近期S.Goswami证明了当自然数集$\mathbb N$被有限着色时，集合$\{a, b, ab, b(a+1)\}$是单色的，这也建立了一个长期存在的Hindman猜想的变体，该猜想询问形式为$\{a, b, ab, a+b\}$的单色集。实际上他推翻了J.Sahasrabudhe提出的猜想，即$\{a, b, a(b + 1)\}$不是分区正则的。在本文中，我们证明了$\{a, b, ab, b(a+1)\}$在零附近是单色的，这意味着对于每一个密集子半群$((0, \infty), +)$的有限着色，集合$\{a, b, ab, b(a+1)\}$是单色的或换句话说，我们会得到$a, b$作为小的密集子半群$((0, \infty), +)$中的元素，使得对于该密集子半群的每一个有限着色，集合$\{a, b, ab, b(a+1)\}$是单色的，此外我们还展示了一个模式$x, y, x+y, \frac{y}{x}$在零附近是分区正则的。
英文摘要：
Recently S. Goswami proved that whenever the set $\mathbb N$ of natural numbers is finitely colored, the set $\{a, b, ab, b(a+1)\}$ is monochromatic which also established a variant of the long-standing Hindman's conjecture, which asks for a monochromatic set of the form $\{a, b, ab, a+b\}$. Actually he disproved a conjecture proposed by J. Sahasrabudhe that $\{a, b, a(b + 1)\}$ is not partition regular. In this paper we prove that $\{a, b, ab, b(a+1)\}$ is monochromatic near zero which means for every finite coloring of a dense subsemigroups of $((0, \infty), +)$, the set $\{a, b, ab, b(a+1)\}$ is monochromatic near zero or in other words, we will get $a, b$ in a dense subsemigroups of $((0, \infty), +)$ as small as we want such that the set $\{a, b, ab, b(a+1)\}$ is monochromatic for every finite coloring of that dense subsemigroups of $((0, \infty), +)$, also we show that the pattern $x, y, x+y, \frac{y}{x}$ is partition regular near zero.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
