AI 中文总结
该研究针对欧几里得空间及整数格的稠密子集，证明了含单形及其重心的构型的密度拉姆齐定理，在特定条件下推广了相关度量下的等距副本存在性结果。
AI 中文摘要
我们获得了由单形Δ_o的顶点及其重心构成的构型的密度拉姆齐定理。证明了任何正上密度的子集A⊆ℝⁿ都包含Δ_o及其重心的所有足够大膨胀的等距副本。由于该构型是非球性的，关于二次欧氏度量无法得到此类结果，故考虑由正定、齐次且次数至少为4的形式定义的一般度量ρ。在离散情形下，对整数格ℤⁿ的子集A，在该集合能包含单形等距副本的尺度λ满足某些自然且必要的同余限制时，我们证明了类似的结果。
英文摘要
We obtain density Ramsey theorems for configurations consisting of the vertices of a simplex $Δ_o$ together with their barycenter. We prove that any subset $A\subseteq\mathbb{R}^n$ of positive upper density contains an isometric copy of all sufficiently large dilates of $Δ_o$ together with its barycenter. As this configuration is non-spherical such results are not possible with respect to the quadratic Euclidean metric, we consider general metrics $ρ$ defined by a positive-definite, homogeneous forms of even degree at least four. We prove the analogous result in the discrete setting, for subsets $A$ of the integer lattice $\mathbb{Z}^n$, under some natural and necessary congruence restrictions on the scales $λ$ at which the set $A$ can contain an isometric copy of the simplex.
