TY - JOUR

T1 - On ordered Ramsey numbers of bounded-degree graphs

AU - Balko, Martin

AU - Jelínek, Vít

AU - Valtr, Pavel

N1 - Funding Information:
The authors were supported by the grant GAČR 14-14179S. The first author acknowledges the support of the Grant Agency of the Charles University, GAUK 690214 and the project SVV-2015-260223. The second author received financial support from the Neuron Foundation for Support of Science.
Funding Information:
The authors were supported by the grant GAČR 14-14179S . The first author acknowledges the support of the Grant Agency of the Charles University , GAUK 690214 and the project SVV-2015-260223. The second author received financial support from the Neuron Foundation for Support of Science .
Publisher Copyright:
© 2018 Elsevier Inc.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - An ordered graph is a pair G=(G,≺) where G is a graph and ≺ is a total ordering of its vertices. The ordered Ramsey number R‾(G) is the minimum number N such that every 2-coloring of the edges of the ordered complete graph on N vertices contains a monochromatic copy of G. We show that for every integer d≥3, almost every d-regular graph G satisfies R‾(G)≥[Formula presented] for every ordering G of G. In particular, there are 3-regular graphs G on n vertices for which the numbers R‾(G) are superlinear in n, regardless of the ordering G of G. This solves a problem of Conlon, Fox, Lee, and Sudakov. On the other hand, we prove that every graph G on n vertices with maximum degree 2 admits an ordering G of G such that R‾(G) is linear in n. We also show that almost every ordered matching M with n vertices and with interval chromatic number two satisfies R‾(M)≥cn2/log2n for some absolute constant c.

AB - An ordered graph is a pair G=(G,≺) where G is a graph and ≺ is a total ordering of its vertices. The ordered Ramsey number R‾(G) is the minimum number N such that every 2-coloring of the edges of the ordered complete graph on N vertices contains a monochromatic copy of G. We show that for every integer d≥3, almost every d-regular graph G satisfies R‾(G)≥[Formula presented] for every ordering G of G. In particular, there are 3-regular graphs G on n vertices for which the numbers R‾(G) are superlinear in n, regardless of the ordering G of G. This solves a problem of Conlon, Fox, Lee, and Sudakov. On the other hand, we prove that every graph G on n vertices with maximum degree 2 admits an ordering G of G such that R‾(G) is linear in n. We also show that almost every ordered matching M with n vertices and with interval chromatic number two satisfies R‾(M)≥cn2/log2n for some absolute constant c.

KW - Bounded degree

KW - Ordered Ramsey number

KW - Ordered graph

KW - Ramsey number

UR - http://www.scopus.com/inward/record.url?scp=85048719370&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2018.06.002

DO - 10.1016/j.jctb.2018.06.002

M3 - Article

AN - SCOPUS:85048719370

SN - 0095-8956

VL - 134

SP - 179

EP - 202

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

ER -