4 problems

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…

Given an integer $n$ and a real number $r\in[0,1]$, let $G(n,r)$ be the random geometric graph obtained by placing $n$ points independently and uniformly in $[0,1]$ and joining pai…

Let $G\sim G(n,p)$ be a binomial random graph, and consider spanning subgraphs obtained by deleting edges so that the resulting graph has minimum degree at least…

Bounded-degree tree universality resilience conjecture. The conclusion of the cited resilience theorem—that the local resilience of $G(n,p)$ with respect to being universal for…