Hindman定理的薄集形式
Thin Set Versions of Hindman's Theorem
摘要：
在本文中，我们研究的反向数学强度的变化Hindman的定理HT构造的HT基本上结合HT与薄集定理TS，以获得一个原则，我们称之为薄HT。thin-HT表示每个着色$c: \mathbb{N} \to \mathbb{N}$都有一个无限集合$S \subseteq \mathbb{N}$，它的有限和对于$c$是瘦的，这意味着对于所有$s \in S$都有一个$i$，它具有$c(s) \neq i$。我们证明了存在一个瘦HT的可计算实例，使得每个解都计算$\emptyset'$，就像HT的情况一样（参见Blass，Hirst，and Simpson 1987）。在分析这一证明时，我们推出瘦HT蕴涵在$RCA_0 + IΣ^0_2$上的$ACA_0$。另一方面，使用Rumyantsev和沈的可计算版本的Lovász局部引理，我们表明，有一个可计算的实例限制薄HT的总和正好2个元素，使任何解决方案有对角不可计算的程度相对于$\emptyset'$。因此，有一个可计算的实例，这种限制薄HT没有$Σ^0_2$解决方案。
英文摘要：
In this paper we examine the reverse mathematical strength of a variation of Hindman's Theorem HT constructed by essentially combining HT with the Thin Set Theorem TS to obtain a principle which we call thin-HT. thin-HT says that every coloring $c: \mathbb{N} \to \mathbb{N}$ has an infinite set $S \subseteq \mathbb{N}$ whose finite sums are thin for $c$, meaning that there is an $i$ with $c(s) \neq i$ for all $s \in S$. We show that there is a computable instance of thin-HT such that every solution computes $\emptyset'$, as is the case with HT (see Blass, Hirst, and Simpson 1987). In analyzing this proof, we deduce that thin-HT implies $ACA_0$ over $RCA_0 + IΣ^0_2$. On the other hand, using Rumyantsev and Shen's computable version of the Lovász Local Lemma, we show that there is a computable instance of the restriction of thin-HT to sums of exactly 2 elements such that any solution has diagonally noncomputable degree relative to $\emptyset'$. Hence there is a computable instance of this restriction of thin-HT with no $Σ^0_2$ solution.
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