Mathematics Problem Archive

Question Is the set of Fibonacci numbers 3-accessible?...

Conjecture An integer partition is wide if and only if it is Latin....

Given integers $k,n\ge2$, let $d(k,n)$ be the smallest integer $d\ge2$ such that the symmetric group $\frak S$ on the set of all words of length $n$ o...

Let $E$ be a non-empty finite set. Given a partition $\bf h$ of $E$, the stabilizer of $\bf h$, denoted $S(\bf h)$, is the group formed by all permuta...

Begin with the generating function for unrestricted partitions: (1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)... Now change some of the plus signs to ...

Conjecture Define a $2 \times n$ array of positive integers where the first row consists of some distinct positive integers arranged in increasing ord...

Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an $n \times n$ grid. The first player (if...

Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an $n \times n$ grid. The first player (if...

Conjecture Every surreal number has a unique sign expansion, i.e. function $f: o\rightarrow \{-, +\}$, where $o$ is some ordinal. This $o$ is the leng...

Let $S_n$ denote the set of all permutations of $[n]=\set{1,2,\ldots,n}$. Let $i(\pi)$ and $d(\pi)$ denote respectively the number of increasing and t...

Conjecture The largest measure of a Lebesgue measurable subset of the unit sphere of $\mathbb{R}^n$ containing no pair of orthogonal vectors is attain...

Question What is the saturation number of cycles of length $2\ell$ in the $d$-dimensional hypercube?...

Problem Determine the smallest percolating set for the $4$-neighbour bootstrap process in the hypercube....

Problem Bound the extremal number of $C_{10}$ in the hypercube....

Conjecture Does there exist a nontrivial perfect 2-error-correcting code over any finite alphabet, other than the ternary Golay code?...

A $(v, k, t)$ covering design, or covering, is a family of $k$-subsets, called blocks, chosen from a $v$-set, such that each $t$-subset is contained i...

Conjecture If ${\mathbb F}$ is a finite field with at least 4 elements and $A$ is an invertible $n \times n$ matrix with entries in ${\mathbb F}$, the...

Conjecture For every prime $p$, there is a constant $c(p)$ (possibly $c(p)=p$ ) so that the union (as multisets) of any $c(p)$ bases of the vector spa...

Conjecture If $A$ is an invertible $n \times n$ matrix, then there is an $n \times n$ submatrix $B$ of $[A A]$ so that $perm(B)$ is nonzero....

Conjecture If $B_1,B_2,\ldots B_p$ are invertible $n \times n$ matrices with entries in ${\mathbb Z}_p$ for a prime $p$, then there is a $n \times (p-...

Let $M$ be a matroid of rank $r$, and for $0 \le i \le r$ let $w_i$ be the number of closed sets of rank $i$. Conjecture $w_0,w_1,\ldots,w_r$ is unim...

Let $G$ be an (additive) abelian group, and for every $S \subseteq G$ let ${\mathit stab}(S) = \{ g \in G: g + S = S \}$. Conjecture Let $M$ be a mat...

Conjecture If $M_1,\ldots,M_k$ are matroids on $E$ and $\sum_{i=1}^k rk_{M_i}(X_i) \ge \ell (k-1)$ for every partition $\{X_1,\ldots,X_k\}$ of $E$, th...

Question Is the binary affine cube $AG(3,2)$ the only 3-connected matroid for which equality holds in the bound $$E(M) \leq c(M) c(M^*) / 2$$where$c(M...

Conjecture Let $r \ge 2$ be an integer and let $H$ be a minor minimal clutter with $\frac{1}{r}\tau_r(H) < \tau(H)$. Then either $H$ has a $J_k$ minor...

Question Does there exist a constant $c>1/2$ and a function $n_0(k)$ such that if $|X|\geq n_0(k)$, then every saturated $k$-Sperner system $\mathcal{...

Let $R(k,k)$ denote the $k^{th}$ diagonal Ramsey number. Conjecture $\lim_{k \rightarrow \infty} R(k,k) ^{\frac{1}{k}}$ exists. Problem Determine th...

Conjecture If $A$ is 2-large, then $A$ is large....

For $k$ and $\ell$ positive integers, the (mixed) van der Waerden number $w(k,\ell)$ is the least positive integer $n$ such that every (red-blue)-colo...

We let $Q_d$ denote the $d$-dimensional cube graph. A map $\phi: E(Q_d) \rightarrow \{0,1\}$ is called edge-antipodal if $\phi(e) \neq \phi(e')$ whene...

The Betti poset of a monomial ideal $I$ determines the Stanley projective dimension of $S/I$ and $I$. More precisely, if $I\subseteq S$ and $I'\subset...

Given a graph $G(V,E)$, find a minimum cardinality set $S \subset V$ of nodes such that adding a complete graph on $S$ renders the graph $G+K_S$ globa...

Problem 1. Given an undirected graph $\displaystyle G=(V,E)$ and $\displaystyle s,t\in V,\;\; k\in N$, decide whether the graph has an acyclic orienta...

Is it true that every t-perfect graph is strongly t-perfect?...

Are there deletion-contraction formulas for the polymatroid Tutte polynomial?...

Let M be a matroid on ground set V, let H=(V,E) be a hypergraph with maximum degree $\Delta$, let c(v) be the cost of node v, and let $l(e) \leq u(e)$...

Let D=(V,A) be a digraph with arc-capacities $c : A \to \mathbb{N}$ and a root node $r_0\in V$. A k-arborescence is the arc-disjoint union of k spanni...

Let G=(A,B;E) be a bipartite graph, and $w:E \to \{1,-1\}$ a conservative weighting. Can we determine in polynomial time the maximum number of positiv...

Is every t-perfect graph 4-colourable?...

A hypergraph $H=(V,E)$ has the Kőnig-property if there is a set $\mathcal{D}\subseteq E$ of pairwise disjoint hyperedges such that there is a vertex c...

If G is an undirected graph with even degrees then call two closed Eulerian walks compatible if no pair of incident edges occurs consecutively in both...

Is there a polynomial algorithm for computing a $v_0$-reduced divisor equivalent to a given divisor of an undirected multigraph?...

Let $G$ be an undirected graph. What is the complexity of computing the rotor-router action of the sandpile group of $G$ on the spanning trees of $G$?...

Is the chip-firing reachability problem co-NP-hard for general digraphs?...

Is the chip-firing halting problem in P for Eulerian digraphs with multiple edges?...

Is it true that the chip-firing halting problem for simple digraphs is NP-complete?...

Is it true that a clutter has the MFMC property if and only if it has the packing property?...

Find a constructive characterization of k-dumpy graphs....

Given a crossing supermodular function $p:2^V\to \mathbb{Z}$ satisfying $p(\emptyset)=p(V)=0$, what is the minimum number of edges of an undirected gr...

Given a crossing supermodular function $p:2^V\to \mathbb{Z}$ satisfying $p(\emptyset)=p(V)=0$, what is the minimum number of pairwise non-parallel arc...