467 problems

Conjecture. There exist an absolute constant $c\ge 1$ and an online algorithm such that, for every finite matroid $M$ and every choice of nonnegative weights, the algorithm selects…

Problem. Is every finite uniformly dense matroid of rank $3$ cyclically orderable?

Let $M=(S,\mathcal{B})$ be a matroid with rank function $r_M$. It is uniformly dense if … for every $X\subseteq S$. A matroid is cyclically orderable if its ground set has a cyclic…

Let $C\in\mathbb{C}^{d\times d}$ be a scaling matrix of rank four. A principal $4$-minor is non-vanishing when the corresponding principal submatrix has nonzero determinant. Rank-f…

For every matroid $M$, the Ehrhart $h^*$-polynomial $h^*_{P_M}(t)$ of its matroid base polytope $P_M$ has only real zeros.

For every linear space $L\subseteq\mathbb{P}^n$, determine the factorization of its principal matroid determinant $E_L$ into irreducible polynomial factors, including the multiplic…

For every matroid $M$, its $g$-polynomial has nonnegative integer coefficients; equivalently, $g_M(t)\in\mathbb{Z}_{\geq 0}[t]$.

Let $\mathrm{SP}$ be the class of series-parallel posets, and let $T(P;x,y)$ denote the Tutte polynomial of the greedoid induced by a poset $P$. The Gordon–McMahon conjecture asser…

For every matroid $M=(E,\mathcal{I})$ and every nonnegative weight function $w:E\to\mathbb{R}_{\ge 0}$, when the elements of $E$ arrive in a uniformly random order and their weight…

For every simple connected graph $G$ on $n$ vertices, let $F(G)$ denote the number of spanning forests of $G$ and let $T(G)$ denote the number of spanning trees of $G$. Then…

For every finite matroid $M$ of rank $r$, let $\Delta(M)$ be its independence complex and let $h(M)=(h_0,\ldots,h_r)$ be defined by…

For every pair of uniform matroids $U_{r,n}$ and $U_{s,m}$, there exists a freest matroid product of $U_{r,n}$ and $U_{s,m}$; equivalently, the relevant class of products has a max…

Does there exist a matroid $N$ that is a second symmetric power of the Vámos matroid $V_8?$ Equivalently, does $V_8$ have a second symmetric power?

Given a directed graph $G$, vertices $s,t\in V(G)$, and an integer $f\ge 1$, construct a sensitivity oracle that, after preprocessing $G$, answers exactly for every set $U$ of at m…

For every bridgeless graph $G$, if every zero of its flow polynomial $F(G,x)$ is real, then $G$ is the dual of a chordal plane graph, and every zero of $F(G,x)$ belongs to the set…

For a matroid $M$, a circuit-cocircuit intersection (CCI) is a subset of its ground set that is the intersection of a circuit and a cocircuit. Oxley's circuit-cocircuit intersectio…

Line-closed matroid–quadratic Orlik–Solomon conjecture. $G$ is line-closed if and only if $A(G)$ is quadratic.

Let $\mathcal{L}$ be a geometric lattice, and let $\mathrm{OS}(\mathcal{L})$ be its Orlik–Solomon algebra. Orlik–Solomon Koszulness conjecture. The algebra…

A positive coline of a matroid is a coline whose copoint partition has more singular classes than multiple classes. A gammoid is a matroid arising from a directed graph by the gamm…

Let $G$ be a connected graph. Its density is … The graph $G$ is uniformly dense if $d(H)\leq d(G)$ for every connected subgraph $H$ of $G$. A graph is cyclically orderable if it ha…

Kajitani–Miyano–Ueno conjecture. The matroid $M$ is cyclically orderable if and only if

A rank-$r$ matroid is paving if all its circuits have cardinality at least $r$. For each positive integer $n$, consider matroids on an $n$-element ground set. The paving matroid co…

Cocircuit-rank conjecture. One has

Let $M$ be a matroid, or let $\mathcal{A}$ be a hyperplane arrangement, and let $A(M)$, respectively $A(\mathcal{A})$, denote its Orlik–Solomon algebra. A matroid or arrangement is…

Let ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ be central hyperplane arrangements defined over the same field. Their intersection lattices $L({\mathcal{A}}_1)$ and…