Lynn Scow: I-indexed Indiscernible Sets and Trees

28/November/2012, 11:o0–12:00
Fields institute,Room 230

Speaker: Lynn Scow (UIC)

Title: $\mathcal{I}$-indexed Indiscernible Sets and Trees

Abstract: Fix any $L’$-structure $\mathcal{I}$ on an underlying set $I$. An $\mathcal{I}$-indexed indiscernible set is a set of parameters $A = \{a_i : i
\in I\}$ where the $a_i$ are same-length finite tuples from some structure $M$ and $A$ satisfies a homogeneity condition: $\textrm{tp}(a_{i_1}, \ldots, a_{i_n};M)=\textrm{tp}(a_{j_1}, \ldots,a_{j_n};M)$ provided that $\textrm{qftp}(i_1,\ldots,i_n;{I})=\textrm{qftp}(j_1,\ldots,j_n;{I})$, where $\textrm{qftp}$ denotes the quantifier-free type. $\mathcal{I}$-indexed indiscernible sets were introduced by Shelah in the 70′s and have important applications in model theory.

In this talk, I will dicuss well-known examples of trees $\mathcal{I}$ for which $\mathcal{I}$-indexed indiscernible sets are particularly well-behaved.
In particular, we will look at the structure $\mathcal{I}_t =
(\omega^{<\omega},\unlhd,\le,\wedge)$ where $\unlhd$ is the partial order on the tree, $\wedge$ is the meet in this order, and $\le$ is the lexicographical order.  Takeuchi and Tsuboi proved that $\mathcal{I}_t$-indexed indiscernibles have a certain technical property, the modeling property.  By a dictionary theorem that I will present in this talk, we may conclude that age($\mathcal{I}_t$) is a Ramsey class.

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