论文标题
邻近的Hindman定理无数群体
The adjacent Hindman's theorem for uncountable groups
论文作者
论文摘要
Hindman，Leader和Strauss以及第二作者和Rinot的最新结果表明，Hindman定理的一些自然类似物因所有不可数的红衣主教而失败。第一作者Komjáth获得了积极方向的结果，第二作者和Lee表明，任意大型的Abelian群体满足了一些Hindman型财产。受到第一作者在可数的环境中研究的类似结果的启发，我们证明了辛德曼的无数枢机主教定理的新变体，称为邻近的欣德曼定理：每一个$κ$都会有一个$λ$，这样，每当$λ$ $κ$κ$κ上的$κ$κ$κ$κ$κ$κ上的$κ$κ时， $ g $搭配所有相同颜色序列相邻术语的有限产品。我们获得$λ$的界限，这是$κ$的函数，并证明了这种界限是最佳的。这是不可数的红衣主教的第一个示例，我们也可以在非亚伯式环境中证明，此外，这是第一个这样的示例，即保证无界长度的单色单色产品（或总和）。
Recent results of Hindman, Leader and Strauss and of the second author and Rinot showed that some natural analogs of Hindman's Theorem fail for all uncountable cardinals. Results in the positive direction were obtained by Komjáth, the first author, and the second author and Lee, who showed that there are arbitrarily large Abelian groups satisfying some Hindman-type property. Inspired by an analogous result studied by the first author in the countable setting, we prove a new variant of Hindman's Theorem for uncountable cardinals, called the Adjacent Hindman's Theorem: For every $κ$ there is a $λ$ such that, whenever a group $G$ of cardinality $λ$ is coloured with $κ$ colours, there exists a $λ$-sized injective sequence of elements of $G$ with all finite products of adjacent terms of the sequence of the same colour. We obtain bounds on $λ$ as a function of $κ$, and prove that such bounds are optimal. This is the first example of a Hindman-type result for uncountable cardinals that we can prove also in the non-Abelian setting and, furthermore, it is the first such example where monochromatic products (or sums) of unbounded length are guaranteed.
