17 problems

Let $n$ be a positive integer and let $k$ satisfy … Consider the number of sparse paving matroids of rank $k$ on an $n$-element ground set and the total number of matroids on $n$ e…

Let elementary split matroids and sparse paving matroids be the two classes of matroids described in the source. Ferroni–Schröter conjecture. The class of elementary split matroids…

A matroid is subsequence-interchangeably base orderable (SIBO) if every pair of bases admits the subsequence exchange orderings specified in the paper. Sparse-paving SIBO conjectur…

Let $M$ be a connected matroid of rank $r$ on $[n]$. Write $U_{r,n}$ for the uniform matroid of rank $r$ on $[n]$, let $d_\lambda(M)$ denote the Schubert coefficient indexed by a p…

A matroid $M \subseteq [0,1]^E$ of rank $m$ is sparse paving if, for all $x,y\in\Delta(m,E)\setminus M$, the coordinatewise minimum $z=\min(x,y)$ satisfies $z(E)\leq m-2$. Asymptot…

Sparse paving enumeration conjecture. Sparse paving matroids are conjectured to dominate the enumeration of matroids in general, in the sense that they form the asymptotically domi…

A sparse paving matroid is a matroid whose circuits of size equal to its rank are also hyperplanes. A class of matroids is asymptotically large if the proportion of matroids in the…

For each positive integer $n$, let $\mathsf{S}_n$ be the sparse paving matroid on $[n]$ of rank $\lfloor n/2\rfloor$ with the maximal possible number of circuit-hyperplanes and lex…

Representability conjecture for K(r,t). Every matroid obtained from $K(r,t)$ by relaxing a circuit-hyperplane into a basis is representable.

A matroid stack delta-matroid is a delta-matroid whose proper set systems in its stack are matroids. It is sparse paving when every matroid in its stack is sparse paving, meaning t…

A matroid is sparse paving if it is paving and its dual is paving. Let $sp(n)$ be the number of sparse paving matroids with $n$ elements, and let $m(n)$ be the number of matroids w…

Let $M$ be a matroid on the ground set $[n]$. A matroid is sparse paving if every $r$-subset of $[n]$, where $r$ is the rank of $M$, is either a basis or a circuit. Sparse-paving c…

Sparse paving minor conjecture for sparse paving matroids. Let $N$ be a fixed sparse paving matroid. Asymptotically almost every sparse paving matroid has an $N$-minor.

Sparse paving minor conjecture. Let $N$ be a fixed sparse paving matroid. Asymptotically almost every matroid has an $N$-minor.

Exponential upper bound conjecture. There is an $\epsilon>0$ and a class of matroids $\mathcal{N}\subseteq\mathbb{M}$ containing almost all matroids, such that

Sparse paving conjecture. As $n\rightarrow\infty$, asymptotically almost all matroids on $n$ elements are sparse paving.

Asymptotic base-orderability conjecture. Almost all matroids are base-orderable, in the sense that the proportion of matroids that are base-orderable tends to $1$ asymptotically.