45 problems

Let $\ell=2$, i.e., $x=(x_1,x_2)$, and let $d\in\mathbb N_0$. Suppose $p\in\mathbb R[x,y]$ and $q\in\mathbb R[x,z]$ are real zero polynomials of degree at most $d$ such that…

For a binary matroid $M$, let $\Lambda(M)$ be the minimum, over bases $B$ of its cocycle space, of the maximum size of a cocircuit in $B$. Let $C(M,t)$ denote its characteristic po…

Let $M_1,\ldots,M_n$ be matroids on the same finite ground set $E$, with $\chi(M_i)\leq n$ for every $i=1,\ldots,n$. Suppose $n$ agents have additive valuations $v_1,\ldots,v_n$ of…

Let $M=(E,\mathcal{I})$ be a matroid with rank function $r$. It is Hamiltonian if some cyclic ordering of $E$ has the property that every segment of $r(M)$ consecutive elements is…

Let $M=(E,\mathcal{I})$ be a matroid on the finite set of goods $E$, and let $\chi(M)$ be the least number of independent sets in a partition of $E$. Suppose $\chi(M)\leq n$ and th…

Let $M=(E,\mathcal{I})$ be an independence structure, where $E$ is a finite set and $\mathcal{I}$ is closed under taking subsets. Let $\chi(M)$ be the least number of independent s…

Let $\mathcal{L}$ be a geometric lattice, let $a$ be an atom of $\mathcal{L}$, and let $\underline{\mathrm{H}}_{[a,\widehat{1}]}$ and $\mathrm{H}_{[a,\widehat{1}]}$ denote the augm…

Let $\mathrm{M}$ be a matroid, let $\mathcal{L}(\mathrm{M})$ be its lattice of flats, and let $\Delta(\widehat{\mathcal{L}}(\mathrm{M}))$ be the order complex of its proper nonempt…

Let $\mathrm{M}$ be a matroid, and let its Chow polynomial be the polynomial invariant associated with its lattice of flats. Ferroni–Schröter's conjecture. The Chow polynomial of…

Let $\mathcal{M}$ be a matroid and let … be two bases. Gabow's conjecture. There exist permutations $\sigma$ and $\sigma'$ such that in the sequence … every set of $r$ cyclically c…

Let $\mathcal{M}=(E,\mathcal{I})$ be a matroid whose ground set $E$ can be partitioned into $k$ disjoint bases, and let $S_1,S_2,\ldots,S_\ell$ be pairwise disjoint subsets of $E$.…

Let $M$ be a loopless matroid of rank $r$, let $G$ be a group of automorphisms of $M$, and let … be its complexified matroid Chow ring, with each graded piece carrying the induced…

Let $(G,c)$ be a colored graph, let $T_{c,j}(G)$ denote the family of transversal edge sets associated with color $j$, and let $H$ be a minimally rigid graph. Transversal character…

Weak real zero amalgamation conjecture. There should be a real zero polynomial $Q\in\mathbb{R}[x_i\mid i\in E']$ such that, for $k=1,2$,

A quaternionic unimodular matroid is a matroid representable by a quaternionic unimodular matrix. A matroid has the half-plane property when its bases-generating polynomial is stab…

A matroid has the weak half-plane property when its set of bases agrees with the support of a stable polynomial. A relaxation of a matroid is obtained by relaxing a circuit-hyperpl…

Schweighofer–Sawall's weak real zero amalgamation conjecture. There should be a real zero polynomial $H\in\mathbb{R}[x_1,x_2,y_1,\ldots,y_m,z_1,\ldots,z_n]$ such that

Let $v$ and $w$ be scalings for $\mathbb P^{m-1}\times\mathbb P^{n-1}$, and define … Let $M_v$ and $M_w$ be the linear matroids defined by the columns of $\widehat v$ and…

Let $G_1$ and $G_2$ be complete digraphs on $p$ nodes, with all edge directions agreeing except for the edge between nodes $p-1$ and $p$. Complete-digraph edge-reversal conjecture.…

Let $M=(X,\mathcal{I})$ be a matroid, and let its bases be its maximal independent sets. The base-exchange walk is the random walk on the bases of $M$ that removes a uniformly chos…

Let $L^{(n)}$ be a sequence of primitive geometric lattices, meaning geometric lattices that cannot be written as products of smaller geometric lattices, and suppose that … Let…

Left LBS-tree edge-activity conjecture. For $0\leq k\leq n-1$, the number of left LBS trees with $k$ edges of the form $(i+1,i)$ is equal to the number of left LBS trees with $k$ e…

Let $f\in\mathbb{R}[x_1,\ldots,x_m]$ be multiaffine in $x_1,\ldots,x_n$, with nonzero coefficient of $x_1\cdots x_n$. For indices $i,j$, let $\Delta_{ij}(f)$ denote the correspondi…

Let $k$-triangulations be the maximal graphs on $[n]$ with no $(k+1)$-crossing, and let the generic bar-and-joint rigidity matroid of dimension $2k$ be the matroid obtained from ge…

Stressed-hyperplane relaxation conjecture. If $M$ is Ehrhart positive and $H$ is a stressed hyperplane in $M$, then $\operatorname{Rel}_H(M)$ is also Ehrhart positive.