6 research outputs found
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 R_e(G) of an edge-ordered graph G is the minimum positive integer N such that there exists an edge-ordered complete graph K_N on N vertices such that every 2-coloring of the edges of K_N contains a monochromatic copy of G as an edge-ordered subgraph of K_N.
We prove that the edge-ordered Ramsey number R_e(G) is finite for every edge-ordered graph G and we obtain better estimates for special classes of edge-ordered graphs.
In particular, we prove R_e(G) <= 2^{O(n^3\log{n})} for every bipartite edge-ordered graph G on n vertices.
We also introduce a natural class of edge-orderings, called \emph{lexicographic edge-orderings}, for which we can prove much better upper bounds on the corresponding edge-ordered Ramsey numbers
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