27 problems

Projective-geometry embedding conjecture. There exists an integer $\ell\geq3$ such that every ideal tangled clutter embeds one of $PG(0,2),\ldots,PG(\ell-1,2)$.

Villarreal's determinant conjecture. If $\alpha_0(\mathcal C')=\beta_1(\mathcal C')$ for all minors $\mathcal C'$ of $\mathcal C$ and $x^{v_1},\dots,x^{v_q}$ have degree $d\geq 2$,…

Algebraic Conforti–Cornuéjols conjecture. If

Let $A$ be a $\{0,1\}$-matrix whose columns each contain the same number $d\geq 2$ of $1$'s, with column vectors $v_1,\ldots,v_q$. Let $\mathcal{C}$ be the clutter associated with…

Let $S\subseteq\{0,1\}^{[d]}$ be a strictly polar set-system, meaning that every restriction of $S$ is polar. Let $\operatorname{cuboid}(S)$ denote the cuboid associated with $S$.…

Let $\mathcal{C}$ be an ideal $\tau$-cover-minimal clutter over ground set $V$. Let $Q(b(\mathcal{C}))$ be the associated polyhedron, let $F$ be the minimal face of…

The Dyadic Conjecture. If $\mathcal C$ is an ideal clutter, then

Packing-property normality conjecture. The Rees algebra $S[Iz]$ is normal.

Algebraic packing conjecture. One has

Conforti--Cornuéjols packing problem. A clutter $\mathcal C$ has the packing property if and only if it has the max-flow min-cut property.

Dijoin-clutter $\tau=2$ conjecture. The weighted digraph $(D,w)$ has a dicut of weight two.

$\tau=2$ conjecture. Every ideal minimally non-packing clutter has a cover of size two.

Cube-ideal set conjecture. There exists an integer $k\geq4$ such that for every cube-ideal set, either all the points agree on a coordinate, or there is a subset of at most $k$ poi…

Bounded chromatic number conjecture. There exists an integer $k\geq 4$ such that every ideal clutter without a member of cardinality at most one has chromatic number at most $k$.

Seymour's quarter-integral packing conjecture. Every ideal clutter $\mathcal{C}$ has a $\frac14$-integral packing of value $\tau(\mathcal{C})$.

Non-idealness conjecture. There exists an integer $k\geq 4$ such that every $k$-wise intersecting clutter is non-ideal.

Linear-quotients ridge-chordality conjecture. If $I_{\overline{\mathcal C}}$ has linear quotients, then ${\mathcal C}$ is ridge-chordal.

Exposed-circuit reformulation of Simon's conjecture. Then ${\mathcal C}$ contains an exposed circuit.

Conforti–Cornuéjols clutter conjecture. If a clutter satisfies the packing property, then it satisfies the integer round-down property.

Let $4\mathfrak{C}_d4$ denote the class of chordal $d$-uniform clutters and let $4\mathfrak{C}'_d4$ denote the class of decomposable $d$-uniform clutters. The inclusion conjecture.…

A point-line incidence matrix of a projective plane is the $0$-$1$ matrix whose rows represent lines, whose columns represent points, and whose entries record incidence. A minimall…

A cubic minimally nonideal matrix is a minimally nonideal matrix whose Lehman core is cubic, and an $n\times n$ matrix is square with $n$ rows and $n$ columns. Cubic mni order conj…

A minimally nonideal matrix is a clutter matrix that is not ideal but whose proper minors are all ideal. Its core is the submatrix consisting of its minimum-weight rows. A non-dege…

Let ${\mathscr C}$ be a clutter of a combinatorial affine plane, and suppose that its blocking number is at least $3$. An ideal solution clutter is an ideal clutter associated with…

Let ${\mathscr C}$ be an ideal minimally non-packing clutter, meaning an ideal clutter that does not have the packing property but whose proper minors have the packing property. It…