Related papers: How special is your Aronszajn tree?

The paper is devoted to a study of certain fixed point properties, and their relatives, in the context of full automorphism groups of countable rooted trees. Namely, we study Serre's property (FA'), also called unsplittability, property…

We construct, for every prime p, a function field K of characteristic p and an ordinary abelian variety A over K, with no isotrivial factors, that admits an etale self-isogeny of p-power degree. As a consequence, we deduce that there exist…

Let $k\ge 2$ be an integer and $T_1,\ldots, T_k$ be spanning trees of a graph $G$. If for any pair of vertices $(u,v)$ of $V(G)$, the paths from $u$ to $v$ in each $T_i$, $1\le i\le k$, do not contain common edges and common vertices,…

It is proved that each bounded injective bilateral weighted shift $W$ satisfying the equality $W^{*n}W^{n}=(W^{*}W)^{n}$ for some integer $n\geq 2$ is quasinormal. For any integer $n\geq 2$, an example of a bounded non-quasinormal weighted…

Assuming the existence of a strong cardinal and a measurable cardinal above it, we construct a model of $ZFC$ in which for every singular cardinal $\delta$, $\delta$ is strong limit, $2^\delta=\delta^{+3}$ and the tree property at…

We study strong $r$-central $2$-trees whose non-central vertices have degrees in $\{2,3\}$, focusing on the cases $r=1,2,3$. For each $r$, we derive exact degree constraints relating the maximum degree $\Delta$ to the numbers of degree-$3$…

It is proved that the consistency strength of having definable tree property for successors of all regular cardinals is the consistency strength of having proper class many small large cardinals which are defined very similar to…

We show that on any Riemann surface S of genus g>1 any nonsingular even spin bundle defines e-foloation of S. When a surface is hyperelliptic then all leaves of this foliation are finite and almost all of them consists of 2g+2 points.…

We show that for every graph $H$, there is a hereditary weakly sparse graph class $\mathcal C_H$ of unbounded treewidth such that the $H$-free (i.e., excluding $H$ as an induced subgraph) graphs of $\mathcal C_H$ have bounded treewidth.…

We introduce the split principles and show that they bear tight connections to large cardinal properties such as inaccessibility, weak compactness, subtlety, almost ineffability and ineffability, as well as classical combinatorial objects…

Let $S$ be a Scott set, or even an $\omega$-model of $\mathsf{WWKL}$. Then for each $A\in S$, either there is $X \in S$ that is weakly 2-random relative to $A$, or there is $X\in S$ that is 1-generic relative to $A$. It follows that if…

Let $X$ be a real or complex Banach space and $T_t:X\to X$ is a power bounded operator (or a $C_0$-semigroup). If there exists a "occasionally" attracting compact subset K (for each x$ in unit ball $\liminf_n \rho(T^n x, K)=0$ then there…

We construct a model in which the tree property holds in $\aleph_{\omega + 1}$ and it is destructible under $\text{Col}(\omega, \omega_1)$. On the other hand we discuss some cases in which the tree property is indestructible under small or…

A graph is odd if all of its vertices have odd degrees. In particular, an odd spanning tree in a connected graph is a spanning tree in which all vertices have odd degrees. In this paper we establish a unified technique to enumerate odd…

In this paper we prove that the Scott topology $\mathfrak S$ on a rooted non-metric tree $\mathcal T$ is strictly coarser than the weak tree topology. Moreover, for each $t\in \mathcal T$, we consider a natural order $\preceq_t$ on…

An $\alpha$-thin tree $T$ of a graph $G$ is a spanning tree such that every cut of $G$ has at most an $\alpha$ proportion of its edges in $T$. The Thin Tree Conjecture proposes that there exists a function $f$ such that for any $\alpha >…

For any graph $G$, let $t(G)$ be the number of spanning trees of $G$, $L(G)$ be the line graph of $G$ and for any non-negative integer $r$, $S_r(G)$ be the graph obtained from $G$ by replacing each edge $e$ by a path of length $r+1$…

In this paper we give an exact analytical expression for the number of spanning trees of an infinite family of outerplanar, small-world and self-similar graphs. This number is an important graph invariant related to different topological…

We show that every separable simple tracially approximately divisible $C^*$-algebra has strict comparison, is either purely infinite, or has stable rank one. As a consequence, we show that every (non-unital) finite simple ${\cal Z}$-stable…

A graph is $\alpha$-excellent if every vertex of the graph is contained in some maximum independent set of the graph. In this paper, we present two characterizations of the $\alpha$-excellent $2$-trees.