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 $T \neq$ true arithmetic, then $H(T) = \beth_{\omega1}$. If $\delta \not\rightarrow (\rho)^{<\omega}$, then any completion of Peano Arithmetic has a model of size $\delta$ with no set of indiscernibles of size $\rho$. There are similar results for theories strongly resembling Peano Arithmetic, e.g., $\mathrm{ZF} + V = L$.