Mathematics Problem Archive

Bixby-Flint-Miklos : Let G be a bicolored graph, and let H be the graph whose vertices are the valid pressing sequences of G and whose edges connect t...

Let $A=(a_{ij})$ be an $n\times n$ matrix with nonnegative entries and total entry sum $n$. Define $$\phi(A)=\prod_{i=1}^n\sum_{j=1}^n a_{ij}+\prod_{j...

A superpermutation on $n$ symbols is a string containing every permutation of the $n$ symbols as a contiguous substring. Determine the minimum possibl...

For positive integers $N,q,a$, let $Q(N;q,a)$ be the number of perfect squares among $a,a+q,\ldots,a+(N-1)q$, and let $Q(N)=\max_{q,a\geq1}Q(N;q,a)$. ...

For partitions $\lambda,\mu,\nu$ of $n$, the Kronecker coefficient $g_{\mu\nu}^{\lambda}$ is defined by $$V_\mu\otimes V_\nu\cong\bigoplus_\lambda g_{...

Let $M(n)$ be the number of monotone Boolean functions of $n$ variables, equivalently the number of antichains of subsets of an $n$-element set. Deter...

Let $W(r,k)$ be the least $N$ such that every coloring of $\{1,\ldots,N\}$ with $r$ colors contains a monochromatic arithmetic progression of length $...

For $n\in\mathbb{N}$, let $\Pi(n)$ be the product of its decimal digits, and let $\operatorname{Pm}(n)$ be the least positive integer such that $\Pi^{...

Define $f:\mathbb{N}\to\mathbb{N}$ by $f(n)=n+\operatorname{rev}(n)$, where $\operatorname{rev}$ reverses the decimal digits. Are there infinitely man...

What is the complexity of the union of ``fat'' objects in $\mathbb{R}^3$?...

What is the maximum number of $k$-sets? (Equivalently, what is the maximum complexity of a $k$-level in an arrangement of hyperplanes?)...

What is the complexity of the vertical decomposition of $n$ surfaces in $\mathbb{R}^d$, $d \ge 5$?...

How many intersections among an arrangement of pseudosegments in the plane must be added as vertices to allow the pseudosegment arrangment to be exten...

How many polyominoes on $n$ squares are there? A polyomino is a connected interior-disjoint union of axis-aligned unit squares joined edge-to-edge, in...

For a point set $P$ in $\mathbb{R}^d$, let $f_d(P)$ be the number of unit-distance point pairs: $$f_d(P) = \left| \{ (u,v) \mid u, v \in P, \, \|u-v\|...

For any (planar) triangle $T$, is there is a $3$-coloring of the (infinite) plane with no monochromatic copy of $T$? We imagine congruent copies of $T...

Given a set of $n$ unit-radius balls in $\mathbb{R}^3$, what is the number of lines that are tangent to four of the balls in the set, and miss all the...

Label the faces of a unit cube with numbers $1$--$6$ as in a die. Place the cube to sit on an integer lattice grid, with one corner at the origin and ...

Let us say that two rectangles in the place are independent if both their $x$- and $y$-axis projections are disjoint. A set of rectangles is then inde...