9 problems

Let $C$ be an $n$-vertex oriented cycle, and let $D(n,p)$ be the binomial random digraph with $n$ vertices and edge probability $p$. Ferber–Long conjecture. If … then $D(n,p)$ cont…

Fix $d\geq 2$, and let $H=\ovar{G}(n,d)$ be the sparse random digraph used in the paper. Write $\ovar{r_1}(H)$ for its one-color oriented Ramsey number. Linear random-digraph conje…

Giant strongly connected component size conjecture. Under the assumption of local convergence of $G_n$ in probability, $\frac{1}{n}|SCC_1|$ converges in probability to

Let $D(n,p)$ be the random directed graph and $G(n,p)$ the random graph, with $p=n^{-1}+\lambda n^{-4/3}$. Let $X^\lambda=n^{-2/3}|\mathcal{C}_1(G(n,p))|$ and…

Let $\mathcal{D}$ be a random $k$-regular digraph on $n$ vertices, and let $\rho(\mathcal{D})$ denote its nontrivial spectral radius. Almost-Ramanujan conjecture for random regular…

Spectral-measure conjecture. The whole empirical spectral measure of the adjacency matrix of a $d$-regular digraph converges almost surely in distribution to…

Three-vertex DERD conjecture. Any DERD with $n=3$ is also a VRD.

Let $D\sim\mathcal D(n,p)$ be a random directed graph on vertex set $[n]$, where each possible arc is present independently with probability $p$, and let $C$ be a Hamilton cycle wi…

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