7 problems

Let $P$ be a poset containing no infinite antichain. A chain is a pairwise comparable subset of $P$, and an antichain is a pairwise incomparable subset. Fish-scale conjecture. Ther…

Aharoni–Ziv's matroidal Hall conjecture. There is no $M$-independent base of $N$ if and only if there exists an $X\subseteq E$ such that $M\!\upharpoonright\!X$ has an $N.X$-indepe…

Aharoni–Lovász conjecture. Every perfect graph in which all independent sets are finite is strongly perfect.

Fish bone conjecture. There exist a chain $C$ and a decomposition $D$ of $V(P)$ into disjoint antichains such that every antichain in $D$ meets $C$.

Let $G$ be a graph, and let $\chi(G)$ denote its chromatic number. For $n\in\mathbb{N}$, write $\operatorname{Sh}_n(\omega)$ for the corresponding shift graph on $\omega$. Taylor's…

Let $P$ be a poset of bounded width, viewed through its comparability structure. A chain is a subset of pairwise comparable elements, and an independent set is a subset containing…

Let $G$ be a graph. An independent set is a set of vertices containing no edge of $G$, and a cover of the vertex set is a family of such sets whose union is the vertex set. Strong…