20 problems

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 $d\geq 2$,…

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…

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.

For a $d$-uniform clutter $\mathcal{C}$, let $I(\mathcal{C})$ be its edge ideal, and let $\Omega_{d,k}$ consist of the clutters satisfying … … and having no proper induced sub-clut…

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

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…

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.

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 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…

Let $d$ be a positive integer, and let $\mathscr{C}^{\rm a.tree}_d$, $\mathscr{C}^{\min}_d$, and $\mathscr{C}^{\rm obs}_d$ denote, respectively, the classes of $d$-uniform clutters…

Let $G$ be a perfect graph, and let $\mathcal{C}$ be its clique clutter, whose edges are the maximal cliques of $G$. The clutter $\mathcal{C}$ is ideal when its covering polyhedron…