23 research outputs found
A combinatorial proof of the extension property for partial isometries
We present a short and self-contained proof of the extension property for
partial isometries of the class of all finite metric spaces.Comment: 7 pages, 1 figure. Minor revision. Accepted to Commentationes
Mathematicae Universitatis Carolina
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
Edge-ordered Ramsey numbers
We introduce and study a variant of Ramsey numbers for edge-ordered graphs,
that is, graphs with linearly ordered sets of edges. The edge-ordered Ramsey
number of an edge-ordered graph
is the minimum positive integer such that there exists an edge-ordered
complete graph on vertices such that every 2-coloring of
the edges of contains a monochromatic copy of
as an edge-ordered subgraph of .
We prove that the edge-ordered Ramsey number
is finite for every edge-ordered graph and we obtain better
estimates for special classes of edge-ordered graphs. In particular, we prove
for every bipartite
edge-ordered graph on vertices. We also introduce a natural
class of edge-orderings, called lexicographic edge-orderings, for which we can
prove much better upper bounds on the corresponding edge-ordered Ramsey
numbers.Comment: Minor revision, 16 pages, 1 figure. An extended abstract of this
paper will appeared in the Eurocomb 2019 proceedings in Acta Mathematica
Universitatis Comenianae. The paper has been accepted to the European Journal
of Combinatoric