96 problems

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 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 $\lambda$,

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 $\Phi$ 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 $K_{3(q-1)+1}$ 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.

Cactus-shadow extension conjecture. For any embedded shadow whose block-adjacency graph is a cactus and whose cyclic blocks are annular in the sense of the necklace definition, the…

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…

Let $G$ be a graph, let $\mathcal{U} = \{U_1, \ldots, U_m\}$ be a partition of $V(G)$, and let $k \ge -1$. Write $\mathbf{Col}(\mathcal{I}(G),\mathcal{U})$ for the colorful complex…

Fix $d\geq 2$ and $k\in\{3,\dots,d+1\}$. In the random Borsuk graph process, edges are added in order of decreasing geodesic distance as the angle parameter $\alpha$ increases. For…

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 $T_1$ be a triangulation of the $n$-sphere and let $S_{n+2}^n$ be the boundary triangulation of the $(n+1)$-simplex. For a coloring $L:V(T_1)\to V(S_{n+2}^n)$, let…

Characteristic-zero exterior shifting conjecture.

Barycentric subdivision conjecture. If $K$ is a surface triangulation, then $\Delta^{ex}(\operatorname{sd}(K))$ is a homology lex-segment.

Uniform surface conjecture. For every fixed genus $g$, a.a.s.,

Let $G$ and $H$ be graphs, let $\Omega_{2d+1}(G)$ denote the graph functor used in the source, and let $n$ be a positive integer. Wrochna's graph-index conjecture. If … then for so…

Let $X$ and $Y$ be $\mathbb{Z}_2$-spaces, with product equipped with the diagonal $\mathbb{Z}_2$-action, and let $\operatorname{ind}(X)$ denote the index of a $\mathbb{Z}_2$-space.…