10 problems

A clutter $\mathcal{C}$ packs if $\tau(\mathcal{C},\mathbf{1})=\nu(\mathcal{C},\mathbf{1})$, and it has the packing property if every minor of $\mathcal{C}$ packs. A clutter is min…

Projective-geometry embedding conjecture. Every clean tangled clutter embeds a projective geometry over the two-element field.

Let $\cal C$ be a 2-partitionable clutter. It has the packing property when every minor has the König property, and it is Mengerian when the max-flow min-cut equality holds for eve…

Let $\cal C$ be a clutter with incidence vectors $v_1,\dots,v_q$, and suppose every minor $\cal C'$ satisfies $\tau({\cal C}')=\nu({\cal C}')$. If $x^{v_1},\dots,x^{v_q}$ have degr…

Let $\cal C$ be a clutter, let $\tau({\cal C}')$ and $\nu({\cal C}')$ denote the minimum vertex-cover and maximum matching sizes of a minor $\cal C'$, and let $R[It]$ be the Rees a…

The Cohen–Macaulay conjecture. If ${\mathcal C}$ is an admissible clutter and ${\mathcal C}$ is unmixed, then ${\mathcal C}$ is Cohen–Macaulay.

Let $\mathcal C$ be a uniform clutter with incidence vectors $v_1,\ldots,v_q$. Its max-flow min-cut property means that the associated max-flow and min-cut optimization problems ha…

Let $\mathcal C$ be a uniform clutter with edge incidence vectors $v_1,\ldots,v_q$, and suppose its packing property holds. Define … as the matrix whose columns are the augmented v…

Let $\mathcal C$ be a uniform clutter with incidence vectors $v_1,\ldots,v_q$, and let $P$ be the associated polytope. The clutter satisfies the packing property if every minor sat…

Let $\mathcal{C}$ be an admissible clutter, and let $I(\mathcal{C})$ denote its edge ideal. The clutter is unmixed when all its minimal vertex covers have the same cardinality, and…