Dependence and isolated extensions

In this paper, we show that if $\varphi(x; y)$ is a dependent formula, then all $\varphi$-types $p$ have an extension to a $\varphi$-isolated $\varphi$-type, $p'$. Moreover, we can choose $p'$ to be an elementary $\varphi$-extension of $p$ and $\vert\mathrm{dom}(p') - \mathrm{dom}(p)\vert \le 2 \cdot \mathrm{ID}(\varphi)$. We show that this characterizes $\varphi$ being dependent. Finally, we give some corollaries of this theorem and draw some parallels to the stable setting.