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For a simplicial complex ${\mathcal C}$ denote by $\beta({\mathcal C})$ theminimal number of edges from ${\mathcal C}$ needed to cover the ground set. If${\mathcal C}$ is a matroid then for every partition $A_1, \ldots, A_m$ of theground set there exists a set $S \in {\mathcal C}$ meeting each $A_i$ in atleast $\frac{|A_i|}{\beta({\mathcal C})}$ elements. We conjecture that aslightly weaker result is true for the intersections of two matroids: if${\mathcal D}={\mathcal P} \cap {\mathcal Q}$, where ${\mathcal P},{\mathcalQ}$ are matroids on the same ground set $V$ and $\beta({\mathcal P}),\beta({\mathcal P}) \le k$, then for every partition $A_1, \ldots, A_m$ of theground set there exists a set $S \in {\mathcal D}$ meeting each $A_i$ in atleast $(\frac{1}{k}-\frac{1}{|V|})|A_i|-1$ elements. We prove this for apartition into two sets.

Fair representation in the intersection of two matroids