69 problems

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 $M$ be a matroid and let $P_M$ denote its base polytope. A triangulation of $P_M$ is unimodular if every simplex in the triangulation is unimodular with respect to the lattice…

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 $k=1,2$,

Let $\mathrm{M}$ be a matroid, and define $h_i(\mathrm{M})$ by … where $f_i(\mathrm{M})$ counts independent subsets of size $i$. Dawson's conjecture. For any $i$ and any matroid…

Let $\mathrm{M}$ be a finite matroid, and let $f_i(\mathrm{M})$ denote the number of independent subsets of $\mathrm{M}$ of size $i$. Welsh's conjecture. The sequence…

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 $\mathrm{M}$ 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…

Kotlar–Ziv conjecture. Every $n \times n$ matroidal Latin square has an independent partial transversal of size $n-1$.

Chari's shellable $O$-sequence conjecture. The $h$-vector $h(M)$ of every coloop-free matroid is a shellable $O$-sequence.

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 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…

Initial lex-segment characterization. The hypergraph $H$ is matroidal if and only if its hyperedges form an initial segment with respect to the lexicographic order.

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 $G$ be a graph, and let conditions 1, 2, 3, and 4 be the four conditions in Theorem 1. The graph $G$ is matroidally copious when it satisfies the relevant matroidal condition d…

Mayhew–Newman–Welsh conjecture. Almost all geometric lattices are paving lattices, in the precise sense that

Let $d,n$ be positive integers. Write ${\mathbb P}\operatorname{L}(d,n)_{\boxtimes}$ for the space of square-free Lorentzian polynomials of degree $d$ in $n$ variables modulo posit…

Let $\mathcal{M}$ be a matroid, and let ${\cal X}=(X_1,\ldots,X_k)$ and ${\cal Y}=(Y_1,\ldots,Y_k)$ be two collections of $k$ bases. A symmetric exchange replaces two bases…

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 write … View each graded piece $A^i_{\mathbb{C}}$ as a $\mathbb{C}G$-module, and let…

Canonical realizations conjecture. The set of realizations of $M$ over $\mathbb{E}$ canonically injects into the set of $\hat{H}\mathbb{E}$-module structures on $\hat{H}M$.

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…

An angle arrangement consists of angular constraints on a planar framework, with independence understood in the associated angle matroid. Whiteley and co-authors' conjecture. There…

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…