Past exam of the computer science course of the University of Oxford / 2023 / Combinatorial optimisation / 2 / c / vi / Solution

List $\sigma (Y)\setminus Y=\{x_1,\dots ,x_m\}$ and put $Y_j=Y\cup \{x_1,\dots ,x_j\}$. Each $x_j\in \sigma (Y)$ also belongs to $\sigma (Y_{j-1})$ by (iii). Thus the definition of the closure of a matroid gives $r(Y_j)=r(Y_{j-1})$ at every step. Iteration yieldsIn particular, collecting all elements that individually fail to increase the rank of a matroid does not create a collective increase.