35 research outputs found
EPPA for two-graphs and antipodal metric spaces
We prove that the class of finite two-graphs has the extension property for
partial automorphisms (EPPA, or Hrushovski property), thereby answering a
question of Macpherson. In other words, we show that the class of graphs has
the extension property for switching automorphisms. We present a short,
self-contained, purely combinatorial proof which also proves EPPA for the class
of integer valued antipodal metric spaces of diameter 3, answering a question
of Aranda et al.
The class of two-graphs is an important new example which behaves differently
from all the other known classes with EPPA: Two-graphs do not have the
amalgamation property with automorphisms (APA), their Ramsey expansion has to
add a graph, it is not known if they have coherent EPPA and even EPPA itself
cannot be proved using the Herwig--Lascar theorem.Comment: 14 pages, 3 figure
What does a typical metric space look like?
The collection of all metric spaces on points whose
diameter is at most can naturally be viewed as a compact convex subset of
, known as the metric polytope. In this paper, we
study the metric polytope for large and show that it is close to the cube
in the following two senses.
First, the volume of the polytope is not much larger than that of the cube,
with the following quantitative estimates: Second, when sampling a metric space from
uniformly at random, the minimum distance is at least with high
probability, for some . Our proof is based on entropy techniques. We
discuss alternative approaches to estimating the volume of
using exchangeability, Szemer\'edi's regularity lemma, the hypergraph container
method, and the K\H{o}v\'ari--S\'os--Tur\'an theorem.Comment: 64 pages, 2 figures. v2: Swapped Sections 5 and 6 and added a
reader's guid
Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)
The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..