16 problems

Let $G_n$ be a sequence of finite 3-regular graphs with growing girth, converging locally to the 3-regular tree $T_3$, and assume that the thresholds $p^{G_n}$ remain bounded away…

Falgas-Ravry–Pfenninger giant-component conjecture. For every $p>1/2$, under every model in $\mathcal{D}_{\geq p}(Q_n)$, the hypercube contains a component of size at least

Hofstad–Janson–Łuczak conjecture. $L_1({{\mathbb{CM} }}_{n})$ is concentrated in the barely subcritical regime.

Quenched giant-component conjecture. A quenched version of this theorem holds.

Edge-density variance positivity conjecture. One should have $\sigma^2_\psi>0$ for this edge-density functional and for every non-zero linear combination of $\psi(H):=1$ and…

Let $G(n,\mathbf d_n)$ and $G^*(n,\mathbf d_n)$ be the random graphs with the given degree sequence considered in the paper, and let $\mathcal C_1$ denote their giant component. Th…

Let $R^N_{\mathrm{MR}}$ and $R^N_{\mathrm{NSW}}$ be the sizes of the giant components in MR and NSW configuration-model random graphs with $N$ vertices. Let $D$ have mean $\mu_D$ a…

A GIRG is a geometric inhomogeneous random graph with power-law weight exponent $\tau$ and an edge-density parameter. In the finite-variance regime, $\tau>3$, and when the edge-den…

Let $\mathcal{H}^k(n,p)$ be the random $k$-uniform hypergraph with edge probability $p$, and let $j$-components denote components under the adjacency notion used in the paper. Writ…

Unique giant component implies weak concentration conjecture. Under the background assumption, the unique giant component property implies the weak concentration property.

Edge-transitive graph conjecture. One can always rescale the edge-weight so that the background assumption holds; after such rescaling, both the weak concentration and unique giant…

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…

Critical-window conjecture. The result should hold provided

Asymptotic Gaussianity conjecture. The pair $(\nu(\bar{\tau}),\mu(\bar{\tau}))$ is also asymptotically Gaussian whenever

Let a random hypergraph have $n$ vertices and excess $k=(\lambda-1)n$, where $\lambda>\Lambda$, and suppose it has a component with unbounded excess with high probability. Giant-co…

Moment-condition conjecture. In addition to ConditionC1, it is enough to assume that $D_n^3$ are uniformly integrable, or, equivalently,