101 problems

For every integer $d\ge 1$ and every integer $r\ge 2$, let $[r]$ denote a set of $r$ vertices, and let $K=[r]*[r]*\cdots*[r]$ be the join of $d+1$ copies of $[r]$. For every contin…

Let $\Delta(j,2)$ denote the least integer $d$ such that every collection of $j$ absolutely continuous probability measures on $\mathbb{R}^d$ can be simultaneously equipartitioned…

For every integer $n\geq 2$, if $G_{3\times n}$ is the $3\times n$ grid graph and $M(G_{3\times n})$ is its matching complex, then there exist a finite index set $J$ and integers…

Let $\Delta_{m,n}$ be the chessboard complex, whose faces are the matchings in the complete bipartite graph $K_{m,n}$, with $n\geq m\geq 1$. Let…

For a finite group $G$ and finite free $G$-spaces (or finite free $G$-simplicial complexes) $X$ and $Y$, the generalized topological Hedetniemi conjecture asserts that…

Let $KG^r(n,k)$ be the Kneser hypergraph whose vertices are the $k$-element subsets of $[n]$, and call a subset $S\subseteq[n]$ $r$-stable when any two elements $x,y\in S$ satisfy…

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.

For a graph $G$, let $\text{\tt Hom}(G,H)$ denote its Hom-complex. Let $V_2(\mathbb R^{n-1})$ be the Stiefel manifold of orthonormal $2$-frames in $\mathbb R^{n-1}$. Csorba's conje…

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 $d,r$ 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…

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 $(r-1)n$.

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…

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

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 $K_{3(q-1)+1}$ 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…

Lovász's conjecture. The equation is true for $T=C_{2r+1}$, $r\geq 1$, and all $n$.

Let $\Pi_n$ be the partition lattice, let $S$ be a set of ranks, and let $\Pi_n^S$ denote its rank-selected subposet. Write $\Delta(\Pi_n^S)$ for the order complex of $\Pi_n^S$, le…

Let $G$ be a graph, let $d$ be its maximal valency, and let $K_n$ be the unlooped complete graph on $n$ vertices. For an integer $k\geq -1$, write $\text{Hom}(G,K_n)$ for the Hom c…

Let $d$ be a positive integer, and let ${\mathcal F}$ be a family of at least two pairwise disjoint open convex sets in ${\mathbb R}^d$. A line in ${\mathbb R}^d$ is transversal to…

Hegedüs's conjecture. If $\mathcal{F}\subseteq 2^{[n]}$ is a Sperner family and $|\mathcal{F}|\ge n+1$, then $\mathcal{F}$ is $2$-balanced.