论文标题
具有SDAP $^+$的同质结构的无限维度Ramsey理论
Infinite-dimensional Ramsey theory for homogeneous structures with SDAP$^+$
论文作者
论文摘要
我们证明，对于任何具有有限的关系符号的语言$ \ mathbf {k} $，最多可以满足两个令人满意的sdap $^+$（或lsdap $^+$），$ \ \ \ \ mathbf {k} $，形成Baire Space的子空间的$ \ Mathbf {k} $的子位置是RAM bair seet，bair bair seet ins parse ins parse ins parse set ins parse。满足SDAP $^+$满足的结构包括理由，Rado图和更普遍的，无限制的结构以及通用$ K $ - 分段图，后三种类型具有或不带有额外的致密线性顺序。作为主要定理的推论，我们获得了纳什 - 威廉姆斯定理的类似物，该定理恢复了这些结构的确切大型拉姆西学位，回答了Todorcevic在2019年的Luminy set理论上提出的一个问题。此外，对于理性和类似的均匀结构，我们的方法会产生拓扑拉西空间，从而满足埃伦托克定理的类似物。
We prove that for any homogeneous structure $\mathbf{K}$ in a language with finitely many relation symbols of arity at most two satisfying SDAP$^+$ (or LSDAP$^+$), there are spaces of subcopies of $\mathbf{K}$, forming subspaces of the Baire space, in which all Borel sets are Ramsey. Structures satisfying SDAP$^+$ include the rationals, the Rado graph and more generally, unrestricted structures, and generic $k$-partite graphs, the latter three types with or without an additional dense linear order. As a corollary of the main theorem, we obtain an analogue of the Nash-Williams Theorem which recovers exact big Ramsey degrees for these structures, answering a question raised by Todorcevic at the 2019 Luminy Workshop on Set Theory. Moreover, for the rationals and similar homogeneous structures our methods produce topological Ramsey spaces, thus satisfying analogues of the Ellentuck theorem.
