TY - JOUR

T1 - Edge-ordered Ramsey numbers

AU - Balko, Martin

AU - Vizer, Máté

N1 - Funding Information:
The first author was supported by the grant no. 18-13685Y of the Czech Science Foundation (GAČR) and by the Center for Foundations of Modern Computer Science (Charles University project UNCE/SCI/004). This article is part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 810115).The second author was supported by the Hungarian National Research, Development and Innovation Office – NKFIH under the grant SNN 129364 and KH 130371, by the János Bolyai Research Fellowship of the Hungarian Academy of Sciences and by the New National Excellence Program under the grant number ÚNKP-19-4-BME-287.
Publisher Copyright:
© 2020 Elsevier Ltd

PY - 2020/6/1

Y1 - 2020/6/1

N2 - 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 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 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)≤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.

AB - 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 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 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)≤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.

UR - http://www.scopus.com/inward/record.url?scp=85082483173&partnerID=8YFLogxK

U2 - 10.1016/j.ejc.2020.103100

DO - 10.1016/j.ejc.2020.103100

M3 - Article

AN - SCOPUS:85082483173

SN - 0195-6698

VL - 87

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

M1 - 103100

ER -