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 Re(G) of an edge-ordered graph G
is the minimum positive integer N such that there exists an edge-ordered
complete graph KN on N vertices such that every 2-coloring of
the edges of KN contains a monochromatic copy of G
as an edge-ordered subgraph of KN.
We prove that the edge-ordered Ramsey number Re(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
Re(G)≤2O(n3logn) for every bipartite
edge-ordered graph G on n 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