Direct simplification of the general Carathéodory-bound proof

Develop a more direct proof of the bound on the -triangle-convexity Carathéodory number that avoids the case-dependent bookkeeping and reduces the complexity of the general argument, thereby clarifying the underlying structure of -triangle-convexity.

Background

The paper proves the bound c_(G) t(G)+1 for the Carathéodory number in triangle convexity by a component-counting argument. The proof requires a detailed case analysis, including exceptional components, 4-cycle configurations, and the assignment of triangles to components.

The authors state that the structure is clearest in the extremal case but that they have not been able to simplify the general proof. A more direct proof is identified as a concrete unresolved direction because it could provide additional insight into the structure of triangle convexity.