TY - JOUR

T1 - Edge-ordered ramsey numbers

AU - Balko, M.

AU - Vizer, M.

N1 - Funding Information:
Received June 6, 2019. 2010 Mathematics Subject Classification. Primary 05C55, 05D10. 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). The second author was supported by the Hungarian National Research, Development and Innovation Office – NKFIH under the grant SNN 129364 and KH 130371.
Publisher Copyright:
© 2019, Univerzita Komenskeho. All rights reserved.

PY - 2019/9/2

Y1 - 2019/9/2

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 (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.

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 (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.

UR - http://www.scopus.com/inward/record.url?scp=85073796568&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85073796568

VL - 88

SP - 409

EP - 414

JO - Acta Mathematica Universitatis Comenianae

JF - Acta Mathematica Universitatis Comenianae

SN - 0862-9544

IS - 3

ER -