3 problems

Let $M_1,\ldots,M_n$ be matroids on the same finite ground set $E$, with $\chi(M_i)\leq n$ for every $i=1,\ldots,n$. Suppose $n$ agents have additive valuations $v_1,\ldots,v_n$ of…

Let $M=(E,\mathcal{I})$ be a matroid on the finite set of goods $E$, and let $\chi(M)$ be the least number of independent sets in a partition of $E$. Suppose $\chi(M)\leq n$ and th…

Let $M=(E,\mathcal{I})$ be an independence structure, where $E$ is a finite set and $\mathcal{I}$ is closed under taking subsets. Let $\chi(M)$ be the least number of independent s…