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

We define a C(k) to be a family of k sets F-1, ... , F-k such that conv(F-i boolean OR Fi+1) boolean AND conv(F-j boolean OR Fj+1) = circle divide when {i, i + 1} boolean AND {j, j + 1} = circle divide (indices are taken modulo k). We show that if F is a family of compact, convex sets that does not contain a C(k), then there are k - 2 lines that pierce F. Additionally, we give an example of a family of compact, convex sets that contains no C(k) and cannot be pierced by [k/2] - 1 lines. (c) 2024 Elsevier B.V. All rights reserved.