8 problems

Let $T$ be an infinite tree, and consider the contact process on $T$ started from a single infection. Write $\lambda$ for the infection rate and let the survival probability be the…

Let $T$ be an infinite tree. Call a vertex $v$ essential if $T\setminus\{v\}$ has at least two infinite components. Say that $T$ has strongly exponential growth if there is some in…

Let $T$ be an infinite tree, and let $\rho$ be its root. For the contact process on $T$, define … … … and … where $\overline\mu$ is the upper invariant measure and $\delta_0$ is th…

Let $T$ be a tree. For a vertex $w$ and a neighbor $x$ of $w$, let $T^x$ denote the component containing $x$ after deleting $w$, let $\mu(x)$ denote the number of isomorphism types…

Let $(X,d)$ be an infinite ultrametric space. A labeled ray is a ray equipped with a labeling that generates the indicated ultrametric space. A tree is starlike if it has exactly o…

Let $G$ be a subdegree-finite primitive permutation group with more than one end. Recall that $G$ has property H if, in its faithful action on its structure tree $T$, the pointwise…

Topological Tree Alternative Conjecture. For a given locally finite tree $T$, the number of isomorphism classes of trees that are mutually topological minors with $T$ is either $1$…

A rooted tree $(T,r)$ is a tree $T$ with a distinguished vertex $r$. Two rooted trees $(T,r)$ and $(T',r')$ are twins, or mutually embeddable, when there are injective graph homomo…