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 (Image found) of an edge-ordered graph (Image found) is the minimum positive integer N such that there exists an edge-ordered complete graph (Image found) on N vertices such that every 2-coloring of the edges of (Image found) contains a monochromatic copy of (Image found) as an edge-ordered subgraph of (Image found). We prove that the edge-ordered Ramsey number (Image found) is finite for every edgeordered graph (Image found) and we obtain better estimates for special classes of edge-ordered graphs. In particular, we prove (Image found) 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.
|Number of pages||6|
|Journal||Acta Mathematica Universitatis Comenianae|
|State||Published - 2 Sep 2019|
ASJC Scopus subject areas
- Mathematics (all)