论文标题
通过树木的复合拉姆齐定理
Composite Ramsey theorems via trees
论文作者
论文摘要
我们证明了一个定理，以确保某些拉姆齐家庭的组成仍然是拉姆齐。作为一个应用程序，我们表明，在$ \ mathbb {n} $的任何有限着色中，都有一个无限的套装$ a $，一个与所需的有限套装$ b $一起，带有$（a+b）\ cup（ab）$单色，回答了Kra，Moreira，Richter和Robertson的最新论文中的一个问题。实际上，我们证明了该结果的迭代版本，它还概括了伯格森和莫雷拉的拉姆西定理，该定理以前仅在田野上持有。我们的主要新技术是涉及树而不是序列的颜色聚焦方法的扩展。
We prove a theorem ensuring that the compositions of certain Ramsey families are still Ramsey. As an application, we show that in any finite coloring of $\mathbb{N}$ there is an infinite set $A$ and an as large as desired finite set $B$ with $(A+B)\cup (AB)$ monochromatic, answering a question from a recent paper of Kra, Moreira, Richter, and Robertson. In fact, we prove an iterated version of this result that also generalizes a Ramsey theorem of Bergelson and Moreira that was previously only known to hold for fields. Our main new technique is an extension of the color focusing method that involves trees rather than sequences.
