18 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…

Tyomkyn's conjecture. If a locally finite tree $T$ has a non-surjective embedding, then

Let $M$ be the compact metric space defining the coloring space, let $d$ be the degree of the regular trees, and let $\mathcal{I}_d(M)$ denote the space of invariant probability me…

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 $1$…

Tree Alternative Conjecture. The number of isomorphism classes of trees mutually embeddable with a given tree $T$ is either $1$ or infinite.

Tree Alternative Conjecture. The number of isomorphism classes of trees mutually embeddable with a given tree $T$ is either $1$ or infinite.

Chain-length conjecture. For every ordinal $\alpha<\omega_2$, there is a chain of locally finite trees of length $\alpha$.

Tyomkyn's conjecture. The set $twin(T)$ is infinite, except when $T$ is the one-way infinite path:

Decidability conjecture. Satisfiability for $\mathsf{ECTL}^*$ with constraints over $\mathcal{T}_{\infty}$ is decidable.

Hard-core uniqueness threshold conjecture. The general uniqueness result for the hard-core model on trees should hold under the condition

Let the nested pushdown tree hierarchy be the hierarchy introduced in the paper, and let the higher-order pushdown and collapsible higher-order pushdown hierarchies be the correspo…

Let $n$-NPT denote the class of level $n$ nested pushdown trees, and let level $n$ collapsible pushdown graphs be the corresponding graph class. An interpretation is a logical inte…

Let $T$ be a locally finite tree, and let $m(T)$ denote the number of isomorphism classes of trees mutually embeddable with $T$. A self-embedding of $T$ is an embedding from $T$ in…