30 problems

Critical logarithmic CLT conjecture. Let $d\in\mathbb{N}$. There exists a constant $σ_{d,d/2}^2\in(0,\infty)$ such that

Penrose–Ong conjecture. Suppose $d\in\mathbb{N}$. The limit theorems for $\d835α\in(0,d/4)$, namely the variance limits and the corresponding Gaussian convergence for the binomial…

Let $G(n,d,p)$ be the spherical random geometric graph and let $G(n,p)$ be the Erdős–Rényi random graph. For probability distributions on a finite set, define total variation dista…

For integers $d\geq 1$, let $T_d(n,r)$ be the random graph obtained by sampling $n$ random points of the $d$-dimensional torus $(\mathds{R}/\mathds{Z})^d$ and joining two vertices…

Continuous-circle greedy-routing conjecture. For every $\varepsilon>0$, there exists a constant $K_0=K_0(\varepsilon)$ such that, for all $K\geq K_0$ and sufficiently large $n$, wi…

Spectral detection conjecture. The sharp threshold for distinguishing the RGG from its Erdős–Rényi counterpart is determined by

Let $H_t(\mathcal{G})$ denote the entropy of the random geometric graph distribution at parameter $t$, and let $\bar{p}_{\max}$ be the corresponding edge-probability parameter at w…

Step-isometry extension conjecture. The map $T$ can be extended to a step-isometry on the whole of $X$.

Strong non-Rado conjecture. Typical countable dense sets in $X$ are strongly non-Rado.

Local resilience conjecture. For every $\epsilon\in (0,1/2]$ and integer $d\geq1$, there exists a constant $C=C(d,\epsilon)>0$ such that, for $r\geq C(\log n/n)^{1/d}$, asymptotica…

Matching-size conjecture. Under this assumption,

Let $G_d(n,p)$ be the random geometric graph model, and let $\mathcal{F}$ be a non-trivial monotone graph property, where monotonicity means monotonicity under adding edges. Random…

Let $n,d$ be positive parameters, let $q\in[1,\infty]$, and consider testing … against … Assume $dq^3=\omega(n^3)$. The signed triangle test is the signed subgraph-count test based…

Let $n,d$ be positive parameters, let $q\in[1,\infty]$, and consider testing … against … Here the signed four-cycle test is the signed subgraph-count test based on the four-cycle…

Let $\tau_{p,d}$ be chosen so that the spherical threshold connection…

Let $\Sigma_n$ be the covariance matrix of the length power functionals in the subcritical regime, for $n\geq 2$, with eigenvalues $\lambda_1\leq\ldots\leq\lambda_n$. Distinct-eige…

Let $(a,p,\theta)$ be fixed with $a>1$, and let $T_n^2$ be the two-dimensional torus. In the Bradonjić–Saniee bootstrap percolation model on the random geometric graph $G_{n,r}^2$,…

Let $(a,p,\theta)$ be fixed with $a>1$, and let $\theta_{\mathrm{local}}(a,p)$ denote the threshold for the appearance of large local, symmetrically distributed infectious outbreak…

Let $(a,p,\theta)$ be fixed with $a>1$, and let $G_{n,r}^2$ be the random geometric graph in the Bradonjić–Saniee bootstrap percolation model, with initially infected set $A_0$, fi…

Inclusion-divergence conjecture. Under these assumptions,

Let $C^0$ be a cycle of length $k$ with its signed cycle statistic $\kappa_{C^0}$, and let $\mathcal{G}(n,p,d)$ denote the noisy high-dimensional random geometric graph model witho…

Let $k$ and $p$ be fixed, let $\mathcal{G}(n,p,d)$ denote the noisy high-dimensional random geometric graph model without edge subsampling, and let $\tau_{[k]}$ be the signed $k$-c…

Let $dgeq1$ be an integer, let $\alpha=\alpha(n)\in(0,1/2)$, and let $H$ be an $n$-vertex graph with minimum degree at least $\alpha n$. Write $G^d(n,r)$ for the random geometric g…

Let $G(n,\pi)$ be the unit disk graph, and let two vertices be displaced by $r_{xy}$. Write $\sigma_{r_{xy}}$ for the number of geodesic paths between them. Negative-binomial conje…

Consider the online power-of-choices geometric graph process on $n$ vertices, with radius … Here $c>0$, $f$ is a function from $(0,\infty)$ to $(0,1)$, and w.h.p. means with high p…