5 problems

Let $G(m)$ be the graph after $m$ steps of the Achlioptas process on $n$ vertices, and let $L_1(m)$ denote the size of its largest component. Under the product rule, given two pote…

A bounded-size rule has a constant $B$ such that all component sizes greater than $B$ are treated identically; a bounded-size 2-Achlioptas rule offers two candidate edges at each s…

Consider the product-rule Achlioptas process on the empty graph with $n$ vertices, in which two candidate vertex pairs are sampled at each step and the pair minimizing the product…

Let $G(m)$ be the random graph after $m$ steps of an Achlioptas process, and let $L_1(G)$ denote the number of vertices in its largest component. A phase transition is explosive if…

Let $H$ be a fixed graph, and let $r$ be the parameter of an Achlioptas process. For a subgraph $H'$ of $H$ and a nonnegative integer $s$, define … Let $\theta^*$ be the minimum of…