Two theorems on the existence of indiscernible sequences (Q1086556)

The author studies generalizations of the following fact: Let \(p(x_ 1,x_ 2,...,x_ i,...)\) be a (noncomplete) type in \(\omega\) variables; then there is an indiscernible sequence realizing p if and only if there is a sequence \(<a_ i:i\in \omega >\) such that, for all increasing f, \(<a_{f(i)}:i\in \omega >\) realizes p. In this rather technical paper, a necessary and sufficient condition is given for a type p in a doubly indexed set of variables \(\{x_{i,\alpha}:\) \(i\in \omega\), \(\alpha\in \kappa \}\) to have a realization \(<a_{i,\alpha}:\) \(i\in \omega\), \(\alpha \in \kappa >\) such that for all \(\alpha\) the sequence \(<a_{i,\alpha}:\) \(i\in \omega >\) is indiscernible over \(\{a_{i,\beta}:\) \(i\in \omega\), \(\beta\neq \alpha \}\) or \(\{a_{i,\beta}:\) \(i\in \omega\), \(\beta <\alpha \}\).