讲座内容：

This lecture continues the study of Helly-type theorems by focusing on fractional and colourful extensions in combinatorial geometry. It begins with exercises illustrating the Fractional Helly Theorem, including the sharp relation $\beta=1-(1-\alpha)^{1/(d+1)}$ and examples involving vertical segments in $\mathbb{R}^2$. The lecture then raises the natural question of what a colourful version of the Fractional Helly Theorem should look like, emphasizing the strength of fractional methods. It further presents a new proof of a covering number theorem using Fractional Helly, showing that for a finite point set $X\subset\mathbb{R}^d$ in general position, there exists a point contained in many $d$-simplices generated by $X$. Finally, the session introduces the Colourful Helly Theorem and discusses its relationship with Colourful Carathéodory's Theorem, including an outline of their equivalence and a more general dimensional formulation.