22 problems

Let $d$ be an integer with $d\geq 3$, let $\alpha$ satisfy … and let $\Delta\sim\Delta(n,n^{-\alpha})$ be a random clique complex. Kahle's bouquet of spheres conjecture. With high…

Let $X\sim X(n,p)$ be a random simplicial complex, let $d\geq 3$, and fix … Set $p=n^{-\alpha}$. Bouquet-of-spheres conjecture. With high probability, $X$ is homotopy equivalent to…

Kahle–Newman conjecture. For every finite abelian $p$-group $G$,

Fix integers $d,m$ with $d\geq m\geq 1$, and suppose that $\alpha$ satisfies … Let $\Delta\sim\Delta_m(n,n^{-\alpha})$ be the corresponding multiparameter random simplicial complex…

Positive-characteristic Betti-number conjecture. For every prime $p$, the normalized mod-$p$ first Betti number converges in probability to the same constant as in characteristic z…

Subquadratic generation conjecture. The normalized minimal number of generators converges to zero in probability:

Consider the infinite clique complexes arising from preferential attachment graphs, and suppose that the spheres formed during the growth process eventually become boundaries of ba…

Let $\beta_q^{(T)}$ denote the $q$-th Betti number of the clique complex of a finite preferential attachment graph with $T$ nodes. The mean-normalized Betti number is…

Let $Y_d(n,p_n)$ be the Linial-Meshulam random simplicial complex, containing all simplexes of dimensions at most $d-1$ and each $d$-simplex independently with probability $p_n$. F…

Luria's conjecture. The exact sharp threshold for simple connectivity is

Let $\mathbb{F}$ be a field and let $Delta\tooltip\sim \mathcal{U}(n)$, where $n$ is even. Here $\mathcal{U}(n)$ denotes the uniform random simplicial complex on $[n]$. Top-homolog…

Let $t$ denote the intensity parameter and let $d$ tend to infinity. The Gaussian and Poisson limit theorems above require growth conditions on $t$ because of dimension-dependent f…

Let $\mathcal{K}_n^{(d)}=\{K_n^{(d)}(t)\}_{0\le t\le1}$ be the $d$-Linial–Meshulam complex process, and let … Here $L_k(\mathcal{K}_n^{(d)})$ is the normalized $k$-th lifetime sum…

Single-step shadow-growth conjecture. In a random evolution of simplicial complexes, the ${\mathbb R}$-shadow grows from linear in $n$ to a giant set of order $\Theta(n^{d+1})$ in…

Coefficient-ring conjecture. The same bound should hold for the vanishing of the $d$-th homology with coefficients in every coefficient ring.

Let $Y_d(n,p)$ be the random $d$-dimensional simplicial complex, and let $c_d^*$ be the real number such that $p=c_d^*/n$ is a one-sided sharp threshold for the non-triviality of…

Let $Y_d(n,p)$ denote the Linial–Meshulam random $d$-dimensional simplicial complex, and let $c_2$ be the constant appearing in the homological phase transition. A sequence…

Let $Y_r(n,n^{-\boldsymbol{\alpha}})$ be the random simplicial complex with multi-exponent $\boldsymbol{\alpha}=(\alpha_0,\alpha_1,\dots,\alpha_r)\in\mathbf{R}^{r+1}$. Consider mul…

Coefficient-ring conjecture. The same bound should hold for all coefficient rings.

Let the clique number be the maximum number of vertices in a clique of a random geometric graph, and consider the subcritical regime. Penrose's conjecture. The clique number become…

Let ${\cal F}_{n,d}$ denote the family of $d$-dimensional random simplicial complexes on $n$ vertices, and let $\gamma_d$ be the constant defined in the paper. Threshold conjecture…

Vanishing conjecture. $betak=0$ asymptotically almost surely.