Hindman定理的回归版本

The reverse mathematics of Carlson's theorem for located words

In this article, we give two proofs of Carlson's theorem for located words in~$\mathsf{ACA}^+_0$. The first proof is purely combinatorial, in the style of Towsner's proof of Hindman's theorem. The second uses topological dynamics to show that an iterated version of Hindman's theorem for bounded sums implies Carlson's theorem for located words.

Actions of trees on semigroups, and an infinitary Gowers--Hales--Jewett Ramsey theorem

We introduce the notion of (Ramsey) action of a tree on a (filtered) semigroup. We then prove in this setting a general result providing a common generalization of the infinitary Gowers Ramsey theorem for multiple tetris operations, the infinitary Hales--Jewett theorems (for both located and nonlocated words), and the Farah--Hindman--McLeod Ramsey theorem for layered actions on partial semigroups. We also establish a polynomial version of our main result, recovering the polynomial Milliken--Taylor theorem of Bergelson--Hindman--Williams as a particular case. We present applications of our Rams…

Effectiveness of Hindman's theorem for bounded sums

We consider the strength and effective content of restricted versions of Hindman's Theorem in which the number of colors is specified and the length of the sums has a specified finite bound. Let $\mathsf{HT}^{\leq n}_k$ denote the assertion that for each $k$-coloring $c$ of $\mathbb{N}$ there is an infinite set $X \subseteq \mathbb{N}$ such that all sums $\sum_{x \in F} x$ for $F \subseteq X$ and $0 < |F| \leq n$ have the same color. We prove that there is a computable $2$-coloring $c$ of $\mathbb{N}$ such that there is no infinite computable set $X$ such that all nonempty sums of at most $2$ …

The reverse mathematics of Hindman's theorem for sums of exactly two elements

Hindman's Theorem (HT) states that for every coloring of $\mathbb N$ with finitely many colors, there is an infinite set $H \subseteq \mathbb N$ such that all nonempty sums of distinct elements of $H$ have the same color. The investigation of restricted versions of HT from the computability-theoretic and reverse-mathematical perspectives has been a productive line of research recently. In particular, HT$^{\leqslant n}_k$ is the restriction of HT to sums of at most $n$ many elements, with at most $k$ colors allowed, and HT$^{=n}_k$ is the restriction of HT to sums of \emph{exactly} $n$ many ele…

Hindman 有限和定理及其在代数拓扑化中的应用

Hindman's finite sums theorem and its application to topologizations of algebras

本文概述了Hindman有限和定理及其基于幂等超滤子的证明技术，并利用该定理的多维版本证明了无限polyring的Zariski拓扑总是非离散的，且相关映射构成闭无处稠密子集，最后提出了代数拓扑化的若干开放问题。

Martin猜想在超算术度上的回归函数情形

Martin's conjecture for regressive functions on the hyperarithmetic degrees

本文证明了Martin猜想的一个版本对所有超算术度上的回归函数成立，关键步骤是化归到连续函数情形，回答了Slaman和Steel的问题。

Chang的猜想可能在超紧基数上失败（提交）

Chang's conjecture may fail at supercompact cardinals (submitted)

我们证明了一个修订版本的拉弗的不灭性定理略有改进的经典结果。当$κ$是超紧的时，应用程序产生$(κ^+,κ)\notcc(\aleph\_1,\aleph\_0)$的一致性。实际证明表明，$ω\_1$ -回归Kurepa-树在超紧基数上是一致的，即使${\rm MM}$在所有正则基数上破坏了它们。这个相当矛盾的事实与一般的直觉相矛盾。

Algebra, selections, and additive Ramsey theory

Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Menger's classic covering property. The methods include, in addition to Hurewicz's game theoretic characterization of Menger's property, extensions of the classic idempotent theory in the Stone--Czech compactification of semigroups, and of the more recent theory of selection principles. This provides strong versions of the mentioned celebrated theorems, where the monochromatic substructures are large, be…