16 problems

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 $G$ 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…

Let ${\cal G}_\kappa$ be a graph of cardinality $\kappa$, and let ${\cal T}_\kappa$ be the tree in which every vertex has degree $\kappa$. For a cardinal $\lambda$, the relation ……

Let $\b5$ be an uncountable regular cardinal, and let ${\emptyset^{\b5}}{\cal H}^{\rho}$ denote the hierarchy introduced in the paper. A poset is strongly $\b5$-scattered if it has…

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.

Let (i) and (iii) denote the corresponding statements in Theorem 1. Full-generality conjecture. Implication (iii) $\rightarrow$ (i) holds in its full generality. Theorem 2 yields o…

For each natural number $n\geq 2$, let $\Gamma_n$ be the $n$-uniform hypergraph on $\mathcal{P}(\omega)$ whose $n$-tuples, modulo finite, form a partition of $\omega$. Let…

The cube-in-cube conjecture. If

Erdős–Rado conjecture. The natural number $bfl$ never depends on $ba$.

Minimum-cardinality antichain conjecture. There is a maximal antichain in $\Pi_\kappa$ of cardinality

Minimum-cardinality chain conjecture. It is not possible for $\Pi_{2^\lambda}$ to have a maximal chain of cardinality strictly less than

For $a\ge 1$, $d\ge 1$, and an infinite cardinal $\alpha$ with $\aleph_0\le\alpha\le 2^{\aleph_0}$, let $h_{a,d}(\alpha)$ be the largest cardinal such that every set of $\alpha$ di…

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 $H$ be a hypergraph whose edge sizes are bounded by a finite number. A matching is a set of pairwise disjoint edges, and a cover of the vertex set is a family of edges whose un…