4 problems

Let $a$, $b$, and $c$ be vertices, and let $\mathbf{P}(ab\\|c)$, $\mathbf{P}(abc)$, $\mathbf{P}(ac)$, and $\mathbf{P}(bc)$ denote the corresponding bond-percolation event probabili…

Let $a$, $b$, and $c$ be vertices, and let $\mathbf{P}(ab\\|c)$, $\mathbf{P}(ac\\|b)$, $\mathbf{P}(abc)$, and $\mathbf{P}(a\\|b\\|c)$ denote the corresponding bond-percolation even…

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…