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We define a $C(k)$ to be a family of $k$ sets $F_1,\dots,F_k$ such that$\textrm{conv}(F_i\cup F_{i+1})\cap \textrm{conv}(F_j\cup F_{j+1})=\emptyset$when $\{i,i+1\}\cap \{j,j+1\}=\emptyset$ (indices are taken modulo $k$). Weshow that if $\mathcal{F}$ is a family of compact, convex sets that does notcontain a $C(k)$, then there are $k-2$ lines that pierce $\mathcal{F}$.Additionally, we give an example of a family of compact, convex sets thatcontains no $C(k)$ and cannot be pierced by $\left\lceil \frac{k}{2}\right\rceil -1$ lines.

Piercing families of convex sets in the plane that avoid a certain subfamily with lines