242 problems

Let $G$ be a decorated unicyclic graph with cycle length parameter $g$, let $R_G$ be its curvature operator, and let $w^\ast$ denote the Perron metric. Write … and let $\varphi$ be…

For the Hermite normal form simplices $S_{\boldsymbol a}=\operatorname{conv}(0,e_1,\ldots,e_{d-1},\boldsymbol a)$, where $\boldsymbol a=(a_1,\ldots,a_{d-1},N)$, and…

Let $\chi_{\mathrm{lat}}(\mathbb{R}^n)$ denote the minimum number of colors in a lattice coloring of $\mathbb{R}^n$ such that no two points of the same color are at Euclidean dista…

For every $d$-dimensional Gorenstein polytope $P$ of index $r$, the stringy $E$-polynomial $E_{\mathrm{st}}(P;u,v)$ vanishes if and only if $P$ is thin, equivalently, its local…

For every integer $n\ge 1$ and every finite, nonzero Borel measure $\mu$ on $\mathbb{R}^n$ that is absolutely continuous with respect to Lebesgue measure, there exist $n$ affine hy…

For each positive integer $n$, define $g(n)$ to be the maximum number of facets of a full-dimensional polytope $P\subseteq\mathbb{R}^n$ whose vertices belong to $\{0,1\}^n$. Determ…

For every integer $n\ge 3$ and every set of $n$ distinct lines in the real projective plane $\mathbb{RP}^{2}$ that are not all concurrent, there exist at least three points of…

For integers $d\ge 2$ and $n\ge d+2$, let $F_1,\ldots,F_n\subseteq\mathbb{R}^d$ be sets, each of which is the union of exactly two disjoint, nonempty, closed convex sets. If, for e…

Conjecture 1 (Lei and Bai). If $G$ is a Ricci-flat $5$-regular graph, then $G$ is either isomorphic to $RF_{72}^{5}$ or is a Cartesian product of the Petersen graph, the Triplex gr…

For each positive integer $n$, let $U(n)$ be the maximum, over all sets $P\subseteq\mathbb{R}^2$ with $|P|=n$, of the number of unordered pairs $\{p,q\}\subseteq P$ satisfying…

Does there exist an absolute constant $c>0$ such that, for every finite set $P\subset\mathbb{R}^3$ with $|P|=N$, the number of distinct pairwise distances satisfies…

Keller's conjecture. For all integers $d\geq 1$, there does not exist a faceshare-free tiling of $\mathbb{R}^d$.

Let $T$ be a tile in $\mathbb{Z}^n$. A tiling by translated copies of $T$ is periodic if it is invariant under a nonzero translation of $\mathbb{Z}^n$. Periodic tiling conjecture.…

Let $N$ points be given in the Euclidean plane, and let a unit distance mean a pair of points whose Euclidean distance is $1$. The unit distance conjecture. The number of unit dist…

Let $n$ be an integer, and let a set of points in the plane be in general position, meaning that no three points are collinear. Erdős–Szekeres convex polygon conjecture. Every set…

Ramos's conjecture. The triple $(d,k,m)$ is a solution if and only if

Let $0 \le k < d$ be integers, and let $r_0, \dots, r_k$ be positive integers. Suppose that $P_0, \dots, P_k$ are sets of points in $\mathds{R}^d$ satisfying … for each $i$. Tverbe…

Let $N>1$ and $3\leq d\leq 41$. Consider an arbitrary TS-packing of $N$ unit balls in $\mathbb{E}^d$. Fejes Tóth's sausage conjecture. The volume of their convex hull is at least t…

Dirac's conjecture. There is a constant $c>0$ such that some point of $S$ is incident to at least $c n$ lines determined by $S$.

Let $A$ be a set of $n$ points in $\mathbb R^d$, and let its diameter complex be the simplicial complex whose faces are subsets of $A$ with diameter equal to the diameter of $A$. S…

Big-Line-Big-Clique Conjecture. For all integers $t$ and $\ell$ there is an integer $n$ such that every finite set $P$ of at least $n$ points in the plane either contains $\ell$ co…

Fejes Tóth's conjecture. This energy is maximal when $x_1,\dots,x_{d+1}$ are mutually orthogonal and $x_k=x_{k-(d+1)}$ for every $k$ with $d+1<k\le N$. Equivalently, periodically r…

Let $\mathcal{P}$ be the set of prime numbers, and let $h(S)$ denote the Helly number of a set $S\subseteq\mathbb{R}^2$. De Loera–La Haye–Oliveros–Roldán-Pensado conjecture. … This…

Let $\mathbf{K}\subset\mathbb{E}^d$ be a $1$-unconditionally symmetric cap body, meaning that $\mathbf{K}$ is symmetric about each coordinate hyperplane of $\mathbb{E}^d$, and let…

Weak Dirac conjecture. Every set $\mathcal{P}$ of $n$ non-collinear points in the plane contains a point incident to at least