Title: Ramsey Spanning Trees and their Applications.

Authors: Ittai Abraham, Shiri Chechik, Arnold Filtser, Michael Elkin and Ofer Neiman.

Abstract: 

The \emph{metric Ramsey problem} asks for the largest subset $S$ of a metric space that can be embedded into an ultrametric (more generally into a Hilbert space) 
with a given distortion. Study of this problem was motivated as a non-linear version of Dvoretzky theorem. Mendel and Naor \cite{MN07} devised the so called Ramsey 
Partitions to address this problem, and showed the algorithmic applications of their techniques to approximate distance oracles and ranking problems.

In this paper we study the natural extension of the metric Ramsey problem to graphs, and introduce the notion of \emph{Ramsey Spanning Trees}. 
We ask for the largest subset $S\subseteq V$ of a given graph $G=(V,E)$, such that there exists a spanning tree of $G$ that has small stretch for $S$. 
Applied iteratively, this  provides a small collection of spanning trees, such that each vertex has a tree providing low stretch paths to {\em all other vertices}. 
The union of these trees  serves as a special type of spanner, a {\em tree-padding spanner}. We  use this spanner to devise the first compact stateless 
routing scheme with $O(1)$ routing decision time, and  labels which are much shorter than in all currently existing schemes.


We first revisit the metric Ramsey problem, and provide a new deterministic construction. We prove that for every $k$, any $n$-point metric space has a 
subset $S$ of size at least $n^{1-1/k}$ which embeds into an ultrametric with distortion $8k$. 
We use this result to obtain the state-of-the-art deterministic construction of a distance oracle.
Building on this result, we prove that for every $k$, any $n$-vertex graph $G=(V,E)$ has a subset $S$ of size at least $n^{1-1/k}$, 
and a spanning tree of $G$, that has stretch $O(k \log \log n)$ between any point in $S$ and any point in $V$.