Related papers: Ladder system uniformization on trees I & II

We show that the Hrushovski-\fraisse limit of certain classes of trees lead to strictly superstable theories of various U-ranks. In fact, for each $ \alpha\in\omega+1\backslash\{0\} $ we introduce a strictly superstable theory of U-rank $…

For an ascending HNN-extension $G*_{\psi}$ of a finitely generated abelian group $G$, we study how a synchronization between the geometry of the group and weak periodicity of a configuration in $\mathcal{A}^{G*_{\psi}}$ forces global…

Let ${\mathbf T}_n$ be a uniformly random tree with vertex set $[n]=\{1,\ldots,n\}$, let $\Delta_{{\mathbf T}_n}$ be the largest vertex degree in ${\mathbf T}_n$, and let $\lambda_1({\mathbf T}_n),\ldots,\lambda_n({\mathbf T}_n)$ be the…

Let $G$ be the Cartesian product of a regular tree $T$ and a finite connected transitive graph $H$. It is shown in arXiv:2006.06387 that the Free Uniform Spanning Forest ($\mathsf{FSF}$) of this graph may not be connected, but the…

The color refinement algorithm is mainly known as a heuristic method for graph isomorphism testing. It has surprising but natural characterizations in terms of, for example, homomorphism counts from trees and solutions to a system of linear…

We extend the classical coding of measured $\mathbb R$-trees by continuous excursion-type functions to c\`adl\`ag excursion-type functions through the notion of parametric representations. The main feature of this extension is its…

We investigate the infinite version of the $k$-switch problem of Greenwell and Lov\'asz. Given infinite cardinals ${\kappa}$ and ${\lambda}$, for functions $x,y\in {}^{\lambda}\kappa $ we say that they are totally different if $x(i)\ne…

This paper continues a line of investigation of the Halpern--L\"{a}uchli Theorem at uncountable cardinals. We prove in ZFC that the Halpern--L\"{a}uchli Theorem for one tree of height $\kappa$ holds whenever $\kappa$ is strongly…

We study the maintenance of a $(\Delta+C)$-edge-coloring ($C\ge 1$) in a fully dynamic graph $G$ with maximum degree $\Delta$. We focus on minimizing \emph{recourse} which equals the number of recolored edges per edge updates. We present a…

A graph on $n$ vertices is equitably $k$-colorable if it is $k$-colorable and every color is used either $\left\lfloor n/k \right\rfloor$ or $\left\lceil n/k \right\rceil$ times. Such a problem appears to be considerably harder than vertex…

Let $K$ be any unital commutative $\bQ$-algebra and $W$ any non-empty subset of $\bN^+$. Let $z=(z_1, ..., z_n)$ be commutative or noncommutative free variables and $t$ a formal central parameter. % Denote uniformly by $\kzz$ and $\kttzz$…

Assuming $\rm PFA$, we shall use internally club $\omega_1$-guessing models as side conditions to show that for every tree $T$ of height $\omega_2$ without cofinal branches, there is a proper and $\aleph_2$-preserving forcing notion with…

Connection of the Four Color Theorem (FCT) with some operations on trees is described. L.H. Kauffman's theorem about FCT and vector cross product is discussed. Operation of transplantation on trees linked with the move of brackets according…

We show that the deterministic decision tree complexity of a (partial) function or relation $f$ lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation $f \circ g$ as long as the gadget $g$…

Let $T^*$ be an almost Suslin tree, that is, an Aronszajn tree with no stationary antichains. Krueger introduced a forcing axiom, $\mathrm{PFA}(T^*)$, for the class of proper forcings that preserve that $T^*$ is almost Suslin. He showed…

This paper introduces the concept of domination in the context of colored graphs (where each color assigns a weight to the vertices of its class), termed up-color domination, where a vertex dominating another must be heavier than the other.…

Stanley [9] introduced the chromatic symmetric function ${\bf X}_G$ associated to a simple graph $G$ as a generalization of the chromatic polynomial of $G$. In this paper we present a novel technique to write ${\bf X}_G$ as a linear…

We study spanning trees on Sierpinski graphs (i.e., finite approximations to the Sierpinski gasket) that are chosen uniformly at random. We construct a joint probability space for uniform spanning trees on every finite Sierpinski graph and…

Surface growth models may give rise to unstable growth with mound formation whose tipical linear size L increases in time. In one dimensional systems coarsening is generally driven by an attractive interaction between domain walls or kinks.…

In a generic Grand Unified Theory with a relatively small dispersion of the spectrum around the $Z$-boson and the unification masses, a connection is established, exact at one loop level, between $M_Z$, $G_{\rm F}$, $\alpha(M_Z)$ and the…