34 problems

Connectivity conjecture. If

Cohomology-vanishing conjecture. If

Łuczak–Peled conjecture. For every $d \geq 2$ and $p=p(n)$ such that $|np-c_d|$ is bounded away from $0$, $H_{d-1}(Y_d(n,p))$ is torsion-free asymptotically almost surely.

Let $K\subseteq {\mathbb R}^d$ be a convex body with smooth boundary, and let the distribution be uniform on $K$. Write $r_n$ for the radius parameter and ${{\mathcal R}}(n,r_n)$ f…

Kahle's conjecture. With high probability, $H_k(X(G),\Gamma)=0$. This conjecture concerns the threshold for homological vanishing in random clique complexes; the paper proves a fir…

Let … denote the $k$th Betti number of a random neighborhood or clique complex, with edge probability $p$. Unimodality conjecture. For both random neighborhood and clique complexes…

Let $T_n$ be the random $2$-dimensional determinantal hypertree on $n$ vertices, and let $H_1(T_n,\mathbb{F}_2)$ denote its first homology over the field with two elements. A seque…

Let $T_{n,d}$ be a $d$-dimensional determinantal hypertree on the vertex set $[n]$, distributed according to the determinantal hypertree measure. A simplicial complex is a cosystol…

Let $\Gamma_{n,m,p}$ and $Y_{n,k,q}$ be the distributions considered in the paper, let $z_Kx$ denote the relevant count associated with a positive $k$-pure shape $x$, and set…

Let $\beta=k-\frac{1}{n_m}$, $p=n^{-\frac{1}{\beta}}$, and $q=n^{-\frac{km}{\beta}}$, with $n=n_m$. Let $\Gamma_{n,m,p}$ and $Y_{n,k,q}$ denote the two distributions defined in the…

Let $N_m(G)$ be the simplicial complex whose faces have at least $m$ common neighbors in a graph $G$. For suitable parameter sequences, compare the distribution of the resulting…

Cohomology approximation conjecture. If $d \geq 4(2k+1)^2$, then, as $h$ tends to infinity, the induced map $\tau^*$ on the $k$th cohomology groups is an isomorphism to…

Borsuk–Ulam conjecture. With high probability, for every continuous map $f\colon\widetilde Z\to\mathbb{R}^{2d+1}$, there exists an $x\in|\widetilde Z|$ such that

Sharp-threshold conjecture. For

Let $d \in \mathbb{N}$ and let $(\Delta_n)_{n \in \mathbb{N}}$ be a sequence of finite $d$-dimensional buildings whose thickness satisfies … as $n \rightarrow \infty$. Let…

Let $\Delta_{k,n}$ be the finite spherical building of type $A_k$, let $\mathcal{P}(\Delta_{k,n})$ denote its induced subcomplexes, and let $\mathcal{S}(\Delta_{k,n})$ denote the s…

Let $K\subseteq {\mathbb R}^d$ be a convex body with smooth boundary, and let the distribution be uniform on $K$. Write $r_n$ for the radius parameter and ${{\mathcal R}}(n,r_n)$ f…

Let $Q$ be a random 2-dimensional cubical complex sampled as $Q\sim Q_2(n,p)$, where $p>1-(\tfrac 12)^{1/40}$. The fundamental group $\pi_1(Q)$ is the group associated with $Q$. To…

Let $p_k$ be the probability at which the density of connected components with homotopy type a wedge of $k$ copies of $S^1$ attains its maximum, and let $p_c$ denote the percolatio…

Let $\mathcal{C}_r$ be the random Čech complex generated by a homogeneous Poisson process on the flat torus, let $H_k(\mathcal{C}_r)$ denote its $k$-th homology, and let $\Lambda$…

Let $X(n,p)$ be the random flag complex obtained from the Erdős–Rényi random graph $G(n,p)$ by adding a $k$-simplex for every clique of $k+1$ vertices. For each $k$, the vanishing…

Let $Y_2(n,p)$ be the Linial–Meshulam random $2$-complex on $n$ vertices, where each possible $2$-simplex is included independently with probability $p$, and let $\gamma:=4^4/3^3$.…

Let $G_k^{{\mathcal C},{\mathcal U}}$ denote the relevant $k$-dimensional random Čech complex, let $n$ be the number of sampled points, let $r_n$ be its scale parameter, and let…

Let $c_{m,k}^{\vartheta}$ denote the maximal $\vartheta$-persistent Betti number $\beta_k^{\vartheta}(P_m,r)$ over all point sets $P_m$ of $m=m(\vartheta,k)$ vertices and all…

Let $\mathcal{Y}_d(n)$ be the stochastic Linial–Meshulam process in dimension $d\geq2$. Let $LT(n)$ denote its largest torsion group, and for a $d$-complex $Y$ with complete…