2 problems

Let $m_k^*$ denote the least integer such that every finite family in a given range space admits a colouring with $k$ colours in which every point covered by at least $m_k^*$ range…

Let ${\mathcal T}$ be the family of all translates of a fixed compact convex set in the plane, and let $\widetilde{f}_{\mathcal T}(2)$ be the minimum number such that every finite…