27 problems

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…

Structural conjecture. The class of $3$-connected binary matroids that do not contain $M(K_4)$ as an induced minor is exactly the class of matroids that can be obtained by starting…

Let $M$ be an Eulerian binary matroid, viewed as a subset of $\mathbb F_2^n\setminus\{0\}$. Let $c_2(M)$ be the minimum size of an odd-cover of $M$, and let $a(M)$ be the minimum n…

Let $M$ be an Eulerian binary matroid, meaning a subset of $\mathbb F_2^n\setminus\{0\}$ whose elements can be decomposed into circuits. Write $c(M)$ for the minimum number of circ…

Let $M$ be a simple binary matroid. A triangle packing is a set of pairwise disjoint triangles of $M$, and a triangle hitting set is a set of elements meeting every triangle of $M$…

Binary cocircuit conjecture. For each integer $k\geq 5$, every minimally vertically $k$-connected binary matroid with at least $2k+2$ elements has a $k$-cocircuit.

Let $M$ be a connected binary matroid that is simple or cosimple. The statements (a), (b), and (c) in the degree-five lemma are equivalent. Degree-five characterization conjecture.…

Let $M$ be a connected binary matroid that is simple or cosimple. The statements (a), (b), and (c) in the degree-two lemma are equivalent. Degree-two characterization conjecture. T…

Generalized bounded-critical-number conjecture. For any $1\leq s\leq t$, the simple $I_{s,t}$-free and triangle-free binary matroids have bounded critical number.

Bonamy–Kardoš–Kelly–Nelson–Postle conjecture. For any $s\geq 1$, the simple $I_s$-free and triangle-free binary matroids have bounded critical number.

Bounded critical number conjecture. For any $s\geq 4$, the class of $I_s$-free and triangle-free matroids has bounded critical number.

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

Let $M$ be a binary matroid, let $T_M(x,y)$ denote its Tutte polynomial, and let $z$ be an integer. Las Vergnas's second conjecture. The value … is an odd integer for every binary…

Exponential density conjecture. If $M=(E,G)$ is a full-rank, $r$-dimensional matroid with no induced $I_t$-restriction, then

Let $s$ be a positive integer. A simple binary matroid is a binary matroid with no loops or parallel elements; an induced $I_s$-restriction and an induced $\operatorname{PG}(1,2)$-…

Coordinate-deletion conjecture. If $S$ is a powerful set, then one can find a coordinate such that deleting this coordinate from all the elements of $S$ yields the set…

Coloop-extension conjecture. If $T$ is a powerful set with at least one vector of weight $1$, then $T$ is a coloop extension of some powerful set $S$.

Let $N$ be a binary matroid with critical number $\bchi(N)=c$, and let $\btheta(N)$ denote its critical threshold. Geelen–Nelson's conjecture. … where $i=2$ if and only if no…

Let $N$ be a matroid, let $\chi(N)$ denote its critical number, and let its critical threshold be the infimum of the density thresholds forcing bounded critical number. Geelen and…

Let $N$ be a simple binary matroid of critical number $c\geq 2$. A $k$-codimensional subspace of $N$ is obtained by intersecting its ground set, in a projective-geometry representa…

Let $\mathcal{N}$ be the class of simple binary matroids of critical number $2$. For $\delta\in\{0,\tfrac14,\tfrac12\}$, let $\mathcal{N}_\delta$ be the three subclasses defined in…

Let $M$ be a simple triangle-free binary matroid, with ground-set size $|M|$ and rank $r(M)$. Critical-number strengthening conjecture. If … then $\chi(M)\in\{1,2\}$. The preceding…

Let $M$ be a simple triangle-free binary matroid, let $r(M)$ denote its rank, and let $|M|$ denote its number of elements. The critical number of $M$ is the minimum codimension of…

Graphicness characterization conjecture. The following assertions are equivalent:

Cocircuit-rank conjecture. One has