the third.

3.  None of the main geodesics have common curves.

In all three cases we will find a pants decomposition such that the hierarchy connecting this pants decomposition to each edge in $\alpha\beta\gamma$ is less than $8,900$.

**Case 1**: Assume the main geodesics of all three edges share the curve $v \in {\mathcal{C}}(S)$. Define $v_{\alpha \beta}^{-1}$ to be the curve on $g_{\alpha \beta}$ preceding $v$ and $v_{\alpha \beta}^{+1}$ the curve on $g_{\alpha \beta}$ following $v$ when viewing $g_{\alpha\beta}$ going from $\alpha_0$ to $\beta_0$. Similarly define $v_{\alpha \gamma}^{-1}$, $v_{\alpha \gamma}^{+1}$, $v_{\beta \gamma}^{-1}$, and $v_{\beta \gamma}^{+1}$. See Figure \[Case 1\].

We want to show the geodesics connecting $v_*^{-1}$ to $v_*^{+1}$ in ${\mathcal{C}}(S \backslash v)$ are not too far apart in ${\mathcal{C}}(S \backslash v)$. Connect $v_{\alpha\beta}^{-1}$ to $v_{\alpha\gamma}^{-1}$, $v_{\alpha\gamma}^{+1}$ to $v_{\beta \gamma}^{+1}$ and $v_{\beta\gamma}^{-1}$ to $v_{\alpha\beta}^{+1}$ by geodesics in ${\mathcal{C}}(S \backslash v)$. We now have a loop in ${\mathcal{C}}(S\backslash v)$. Since all curves besides $v$ in $S$ intersect the subsurface $S \backslash v$ non-trivially we can apply the Bounded Geodesic Image Theorem on $[v_{\alpha \beta}^{-1}, \alpha_1]_S$ and $[\alpha_1, v_{\alpha \gamma}^{-1}]_S$ to get $d_{{\mathcal{C}}(S\backslash v)}(v_{\alpha \beta}^{-1}, v_{\alpha \gamma}^{-1}) \leq 2M$. Similarly, $d_{{\mathcal{C}}(S\backslash v)}(v_{\alpha \gamma}^{+1}, v_{\beta \gamma}^{+1}) \leq 2M$ and $d_{{\mathcal{C}}(S\backslash v)}(v_{\beta \gamma}^{-1}, v_{\alpha\beta}^{+1}) 