737 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…

Aharoni–Berger's conjecture.

Let $\mathrm{M}$ be a matroid, and let $\{B_1,B_2\}$ and $\{B_1',B_2'\}$ be unordered pairs of bases such that the multiset union of $B_1$ and $B_2$ is equal to the multiset union…

Mason's rank conjecture. The rank of $N$ is

Let $M$ be a matroid of rank $n$ on a ground set $E$, and let $B_1,\dots,B_n$ be pairwise disjoint bases of $M$. A transversal of $(B_1,\dots,B_n)$ is a set meeting each $B_i$ in e…

Let $\mathbb{F}$ be a finite field. An excluded minor for a class of matroids is a matroid not in the class whose proper minors all belong to the class. Rota's conjecture. The clas…

Let $\Sigma(r,n)$ be the hypersimplex, and consider a subdivision of $\Sigma(r,n)$ into matroid base polytopes. An interior face is a face of the subdivision not contained in the b…

Let $M$ be a matroid, and let … is the vector recording the face-enumeration data of this simplicial complex. A finite sequence is a pure ${\mathcal O}$-sequence if it is the $f$-v…

Athanasiadis–Kalampogia-Evangelinou conjecture. For every matroid $\mathsf{M}$, $h_{\mathsf{M}}(x)$ has only nonpositive real zeros.

Let $M$ be a matroid of rank $r(M)$, with characteristic polynomial … The integers $w_i(M)$ are the Whitney numbers of the first kind. Heron–Rota–Welsh conjecture. The sequence … o…

A matroid $M$ has an $h$-vector $h(M)=(h_0,ots,h_d)$ determined by the face numbers of its independent-set complex. A vector is a pure $O$-sequence if it is the $F$-vector of a pur…

Let $M$ be a matroid of rank $r$ with two disjoint bases $A$ and $B$. A symmetric exchange chooses $e\in A\setminus B$ and $f\in B\setminus A$ such that both $A-e+f$ and $B-f+e$ ar…

Let $\mathsf{M}$ be a matroid, and write its inverse $Z$-polynomial as $Y_{\mathsf{M}}(t)=a_0+a_1t+\cdots+a_nt^n$. A coefficient sequence is unimodal if it increases and then decre…

Merino–Welsh conjecture. Every such graph satisfies

For a matroid $\mathrm{M}$ of rank $r$, let $W_i$ denote the number of flats of rank $i$. Dowling–Wilson's top-heavy conjecture. One has … and $W_i\leq W_{r-i}$ for $i\leq r/2$. Th…

Let $\mathsf{M}$ be a matroid, and write $Q_{\mathsf{M}}(t)=\sum_i a_i t^i$. A sequence is log-concave with no internal zeros if $a_i^2\geq a_{i-1}a_{i+1}$ whenever defined, and it…

Representability conjecture for coordinate q-matroids. If the corresponding matroid of a coordinate $q$-matroid is representable, then the $q$-matroid is representable.

Gedeon's coefficient bound conjecture. The coefficients of the equivariant Kazhdan–Lusztig polynomial of $\mathrm{M}$ are bounded above by those of the equivariant Kazhdan–Lusztig…

Let $M$ be a matroid of rank $r$, and let $(A,B)$ and $(A',B')$ be compatible basis pairs, meaning that … A symmetric exchange is an exchange of elements between the two bases that…

Maximality conjecture.

Whiteley's maximality conjecture. For all $n,d$, ${\mathcal C}^{d-2}_{d-1}(K_n)$ is the unique maximal abstract $d$-rigidity matroid on $K_n$.

Let $n$ be a positive integer, let $\mathbb{K}$ be a field of characteristic zero, and let $f\in\mathbb{K}[x_1,\dots,x_n]$ be a homogeneous polynomial. Set…

Ferroni's real-rootedness conjecture. Both $\operatorname{\underline{H}}_M(x)$ and $\operatorname{H}_M(x)$ should be real-rooted.

Let $G$ be a finite graph and let $\chi_G(x)$ be its chromatic polynomial. Read's conjecture. The absolute values of the coefficients of $\chi_G(x)$ are unimodal. This conjecture w…

Let $\operatorname{{\mathsf M}}$ be a matroid of rank $r+1$, with characteristic-polynomial coefficients…