Models Without Indiscernibles

For T any completion of Peano Arithmetic and for n any positive integer, there is a model of T of size $\beth_n$ with no (n + 1)-length sequence of indiscernibles. Hence the Hanf number for omitting types over T, H(T), is at least $\beth_\omega$. (Now, using an upper bound previously obtained by Julia Knight H (true arithmetic) is exactly $\beth_\omega$). If T ≠ true arithmetic, then $H(T) = \beth_{\omega1}$. If $\delta \not\rightarrow (\rho)^{<\omega}$, then any completion of Peano Arithmetic has a model of size δ with no set of indiscernibles of size ρ. There are similar results for theories strongly resembling Peano Arithmetic, e.g., ZF + V = L.