6 problems

Let $P$ be a path with vertex set $V$, and let $(A_1,\ldots,A_m)$ be a partition of $V$. Simultaneous deficit conjecture. There exists a subset $S\subseteq V$ independent in $P$ su…

Let $\mathcal P$ and $\mathcal Q$ be matroids on the same ground set $V$, let $\mathcal D=\mathcal P\cap\mathcal Q$, and set … A set $S$ represents a part $A$ almost $\alpha$-fairl…

Let $\mathcal P$ and $\mathcal Q$ be matroids on the same ground set $V$, and let $\mathcal D=\mathcal P\cap\mathcal Q$. Suppose $\beta(\mathcal P),\beta(\mathcal Q)\le k$. A parti…

Let $H$ be a graph with maximum degree $\Delta(H)$, and partition its edge set into $E_1,\ldots,E_m$. A matching is a set of pairwise vertex-disjoint edges. Fair matching conjectur…

Let $K_{n,n}$ be the complete bipartite graph with edge set partitioned into $E_1,\ldots,E_m$. For a perfect matching $F$, write $|F\cap E_i|$ for the number of its edges in $E_i$.…

Let $P$ be a path whose vertex set is partitioned into sets $V_1,\ldots,V_m$. An independent set is a set of vertices containing no adjacent pair. Almost fair path representation c…