150 problems

Let $r\geq 2$ be an integer. An almost $r$-embedding is a function from a simplicial complex to $\mathbb{R}^d$ that does not identify $r$ points lying in pairwise disjoint faces. T…

Langerman's conjecture. For any $nd$ masses on $\mathbb{R}^d$, there exist $n$ hyperplanes whose induced chessboard partition is balanced for every one of the $nd$ masses.

Let $G$ be a graph with maximal degree $d$, and let $K_n$ be the complete graph on $n$ vertices. Babson–Kozlov's conjecture. The complex $\operatorname{Hom}(G,K_n)$ is at least…

Let $S^{d-1}$ be a unit sphere in $\mathbb R^d$. Suppose we are given $d$ points $x_1, \ldots, x_d\in S^{d-1}$ and a continuous function $f: S^{d-1}\to \mathbb R$. Knaster's conjec…

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…

Sundaram–Welker's acyclicity conjecture. For $\lambda\vdash\mu$ with $\lambda\neq\mu$, the space $X_{\lambda,\mu}$ is ${\mathbb Q}$-acyclic.

Sundaram–Welker's conjecture. For every number partition $\lambda$,

Let $d,r$ be positive integers and let $k$ be a non-negative integer. A Tverberg $(r+1)$-partition of a finite point set is a partition into $r+1$ parts whose convex hulls have a c…

Georgakopoulos–Haslegrave–Montgomery–Narayanan's conjecture. If $H$ has minimum codegree at least $n/k$, then $H$ contains a spanning copy of a $(k-1)$-sphere.

Let $G=(V,E)$ be a graph, and let $\operatorname{Ind}(G)$ denote its independence complex. Define $\psi$ recursively by … Here $N(e)$ denotes the set of vertices adjacent to an end…

Vega's conjecture. The $k$-matching complex of every caterpillar graph is either contractible or homotopy equivalent to a wedge of spheres.

Binary necklace-splitting conjecture. Given a necklace with $n$ kinds of beads and $r\geq 4$ thieves, there exists a binary necklace splitting of size $(r-1)n$.

Let $\operatorname{Gr}(k,n)$ denote the Grassmannian of $k$-planes in $\mathbb{R}^n$, and let $\operatorname{Gr}_{\geq 0}(k,n)$ be its totally nonnegative part, consisting of the p…

Erickson's conjecture. There are no $4$-polytopes or $3$-spheres on $n$ vertices with $\Omega(n^2)$ non-simplicial facets.

Let a simplicial complex be vertex-homogeneous if its automorphism group acts transitively on its vertices, and call it non-evasive if it can be reduced to a simplex by recursively…

Let $\ell$ and $d$ be positive integers. For an integer $n$, let $X=\{x_1,\ldots,x_\ell\}$ be any set of $\ell$ points on the unit sphere $\mathbb{S}^{n-1}$, and let…

Hom-complex realization conjecture. If

Let $G$ be the complement of the $1$-skeleton of a flag simplicial PL sphere, and let $s$ be the maximal valency of $G$. The associated graph coloring manifolds are the Hom complex…

Let $\Phi$ be a finite crystallographic root system of rank $n$, and let $m$ be a positive integer. The generalized cluster complex $\Delta^m(\Phi)$ is the simplicial complex assoc…

Let $d$ and $q$ be positive integers, and let $\Delta_{(d+1)(q-1)}^{\le d}$ denote the $d$-skeleton of the simplex. A Tverberg partition is a collection of $q$ disjoint faces whose…

Let $d$ and $q$ be positive integers. For a continuous map $f:S^{d-1}\to\mathbb{R}^d$ and a point $p\notin f(S^{d-1})$, define the winding number $W(f,p)\in\mathbb{Z}$ by the homol…

Let $X$ be a zero-dimensional separable metrizable space. The selection property $\mathsf{Split}(\Lambda,\Lambda)$ means that every large open cover of $X$ can be partitioned into…

Let $d$ and $q$ be positive integers, let $\Delta^{(d+1)(q-1)}_d$ be the $d$-skeleton of the $((d+1)(q-1))$-dimensional simplex, and let $f:\Delta^{(d+1)(q-1)}_d\to\mathbb{R}^d$ be…

Björner–Lovász conjecture. If $\operatorname{Hom}(G,H)$ is $k$-connected, then

Let $K_{3(q-1)+1}$ be the complete graph on $3(q-1)+1$ vertices, and consider any drawing of this graph in the plane. For a triangle of edges and a point not on it, its winding num…