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arXiv 1912.08643math.CO
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哈雷-朱厄特数的迭代拉姆齐界

Iterated Ramsey bounds for the Hales-Jewett numbers

Mohammad Golshani, Saharon Shelah

备注：
This paper has been withdrawn by the authors as it has been combined with another work and is now superseded by arXiv:2104.05962
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AI总结：

研究哈雷-朱厄特定理，在限制\(|\eta^{-1}\{\alpha \}|\)符号后证明平行定理，通过连接哈雷-朱厄特数与分界数给出该定理的另一种证明。

摘要：

考虑哈雷-朱厄特定理。其\(k\)维版本表明，在适当假设下，组合空间\(\mathcal{U}_{M, \Lambda} = \{ \eta \mid \eta: M \to \Lambda \}\)存在单色\(k\)维子空间。这里的\(k\)维子空间是指\(M\)有一个划分\(\langle N_0, N_1, \cdots, N_k \rangle\)，\(N_1, \cdots, N_k \neq \emptyset\)（允许\(N_0\)为空）以及某个\(\rho_0: N_0 \to \Lambda\)，子空间由满足\(0<l<k + 1\)时\(\rho \restriction N_l\)为常数且\(\rho \restriction N_0 = \rho_0\)的\(\rho \in \mathcal{U}_{M, \Lambda}\)组成。直观上希望每个\(N_{l}, 0<l<k + 1\)是单元素集，但此时不一定总能找到单色\(k\)维子空间。不过在限制每个\(|\eta^{-1}\{\alpha \}|\)的符号后，我们证明了平行定理，其证明与哈雷-朱厄特定理无关。然后通过表明哈雷-朱厄特数不比分界数大太多来连接这两个数，给出了哈雷-朱厄特定理的另一种证明。

英文摘要：

Consider the Hales-Jewett theorem. The $k$-dimensional version of it tells us that the combinatorial space $\mathcal{U}_{M, Λ} = \{ η\mid η: M \to Λ\}$ has, under suitable assumptions, monochromatic $k$-dimensional subspaces, where by a $k$-dimensional subspace we mean there exist a partition $\langle N_0, N_1, \cdots, N_k \rangle$ of $M$ such that $N_1, \cdots, N_k \neq \emptyset$ (but we allow $N_0$ to be empty) and some $ρ_0: N_0 \to Λ$, such that the subspace consists of those $ρ\in \mathcal{U}_{M, Λ}$ such that for $0<l<k+1, ρ\restriction N_l$ is constant and $ρ\restriction N_0= ρ_0.$ It seems natural to think it is better to have each $N_{l}, 0<l<k+1$ a singleton. However it is then impossible to always find monochromatic $k$-dimensional subspaces (for example color $η$ by $0$ if $|η^{-1}\{α\}|$ is an even number and by $1$ otherwise). But modulo restricting the sign of each $|η^{-1}\{α\}|$, we prove the parallel theorem -- whose proof is not related to the Hales-Jewett theorem. We then connect the two numbers by showing that the Hales-Jewett numbers are not too much above the present ones. This gives an alternative proof of the Hales-Jewett theorem.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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