7 problems

Perfect-matching threshold conjecture. The sharp $d$-threshold for $H_ncup G(n,p)$ to contain a perfect matching coincides with the threshold for $G(n,p)[B]$ to contain a matching…

Let $r$ and $s$ be integers with $1<s<r$, and let $n$ be divisible by $r$. Define … For a graph $G$, let $G^p=G\cup G_{n,p}$ denote its randomly perturbed version, and let a graph…

Directed Hamiltonicity threshold conjecture. The $d$-threshold for Hamiltonicity in randomly perturbed directed graphs is $\eta/n^2$.

Hamiltonicity threshold conjecture. The sharp $d$-threshold for $H_n\cup G(n,p)$ to be Hamiltonian coincides with the sharp threshold for $G(n-d,p)$ to contain a linear forest with…

Perfect-matching threshold conjecture. The sharp $d$-threshold for containing a perfect matching is $C/n$.

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…

Universality conjecture. Asymptotically almost surely, $G_\alpha\cup G(n,p)$ is $\mathcal{F}(n,\Delta)$-universal, meaning that it contains every $n$-vertex graph of maximum degree…