讲座内容：

This lecture continues the study of colourful Helly-type theorems and their connections with convexity and topology. It begins with homework discussions on the binary case of the Colourful Helly Theorem and the Colourful Fractional Helly Theorem. The lecture then focuses on a union version of the Colourful Carathéodory Theorem, showing that if a point lies in the convex hull of every pairwise union $A_i\cup A_j$, then one can choose a colourful transversal whose convex hull still contains that point. The proof is developed through the octahedral construction, where the octahedron $\operatorname{conv}{\pm e_1,\ldots,\pm e_d}$ and its facets play a central role. The lecture also explains why the $\binom{d+1}{2}$ pairwise conditions in the theorem are all necessary. Finally, the session introduces the Colourful Tverberg problem, asking how large the colour classes must be in order to guarantee disjoint rainbow sets with intersecting convex hulls, and concludes by presenting the Borsuk--Ulam Theorem as a key topological tool for such results.