论文标题
超越加尔文 - 辛德曼 - 福斯滕贝格定理：\\分区的规律性在任意半群中
Beyond the Galvin--Hindman--Furstenberg Theorem:\\ Partition regularity of IP sets in arbitrary semigroups
论文作者
论文摘要
根据Furstenberg的说法，SemoGroup中设置的A \ emph {无限平行运输（ip）}是一个包含序列$ a_1，a_2，\ dotsc $的集合，以及所有有限的总和$ a_ {i_1} + \ dotsb + dotsb + dotsb + a_ _ {i_m $ $ $ $ $ $ i_1 $ i_1 Hindman使用Galvin方法证明了自然数的IP集为\ emph {partition Quarric}：对于IP集的每个有限分区，也必须是IP集。 Furstenberg指出，同样的证明也适用于任意的半群。但是，在具有依恋群的半群中，IP集可能是单身人士。 A \ emph {正确的IP Set}是可以选择见证序列$ a_1，a_2，\ dotsc $作为\ emph {Bioctive}（在Naturals中，每个IP集合）。我们提供了半群的完整表征，其中适当的IP集是定期分区的，并表明该属性等同于其他几种添加剂Ramsey理论的基本概念。 使用我们的结果，我们证明，如果对于半群的每个有限着色都有单色适当的IP集，那么对于半群的每种有限着色，都会有无限的多个，成对的脱节，适当的IP集共享相同的颜色。
According to Furstenberg, an \emph{infinite parallelepiped (IP)} set in a semigroup is a set that contains a sequence $a_1,a_2,\dotsc$, together with all finite sums $a_{i_1}+ \dotsb + a_{i_m}$, for natural numbers $m$ and $i_1< \dotsb <i_m$. Using a method of Galvin, Hindman proved that IP sets of natural numbers are \emph{partition regular}: For each finite partition of an IP set, some part must be an IP set, too. Furstenberg noted that the same proof applies to arbitrary semigroups. However, in a semigroup with idempotents, an IP set may be a singleton. A \emph{proper IP set} is one where the witnessing sequence $a_1,a_2,\dotsc$ may be chosen to be \emph{bijective} (in the naturals, every IP set is proper). We provide a complete characterization of semigroups where the proper IP sets are partition regular, and show that this property is equivalent to several other fundamental notions of additive Ramsey theory. Using our results, we prove that if there is a monochromatic proper IP set for each finite coloring of a semigroup, then for each finite coloring of the semigroup there are infinitely many, pairwise disjoint, proper IP sets sharing the same color.
