Edge-ordered Ramsey numbers

Martin Balko; Máté Vizer

doi:10.1016/j.ejc.2020.103100

Edge-ordered Ramsey numbers

Martin Balko, Máté Vizer

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

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(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.

Original language	English
Article number	103100
Journal	European Journal of Combinatorics
Volume	87
DOIs	https://doi.org/10.1016/j.ejc.2020.103100
State	Published - 1 Jun 2020
Externally published	Yes

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics

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10.1016/j.ejc.2020.103100

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title = "Edge-ordered Ramsey numbers",

abstract = "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.",

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AU - Vizer, Máté

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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.

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DO - 10.1016/j.ejc.2020.103100

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VL - 87

JO - European Journal of Combinatorics

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ER -

Edge-ordered Ramsey numbers

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