TY - JOUR

T1 - Ramsey numbers of ordered graphs

AU - Balkoy, Martin

AU - Cibulka, Josef

AU - Králz, Karel

AU - Kynćlx, Jan

N1 - Funding Information:
†Supported by the grants SVV-2013-267313 (Discrete Models and Algorithms), GAUK 1262213 of the Grant Agency of Charles University, and by the project CE-ITI (GACˇR P202/12/G061) of the Czech Science Foundation.
Funding Information:
‡Supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 616787.
Funding Information:
§Supported by the grants SVV-2013-267313 (Discrete Models and Algorithms), GAUK 1262213 of the Grant Agency of Charles University, CE-ITI (GACˇ R P202/12/G061) of the Czech Science Foundation, ERC Advanced Research Grant no 267165 (DISCONV), and by Swiss National Science Foundation Grants 200021-137574 and 200020-144531.
Funding Information:
Supported by the grants SVV-2013-267313 (Discrete Models and Algorithms), GAUK 1262213 of the Grant Agency of Charles University, and by the project CE-ITI (GA?R P202/12/G061) of the Czech Science Foundation. Supported by the European Research Council under the European Union's Seventh Framework Pro-gramme(FP/2007-2013)/ERC Grant Agreement n. 616787. Supported by the grants SVV-2013-267313 (Discrete Models and Algorithms), GAUK 1262213 of the Grant Agency of Charles University, CE-ITI (GA?R P202/12/G061) of the Czech Science Foundation, ERC Advanced Research Grant no 267165 (DISCONV), and by Swiss National Science Foundation Grants 200021-137574 and 200020-144531.
Publisher Copyright:
© 2020, Australian National University. All rights reserved.

PY - 2020/1/1

Y1 - 2020/1/1

N2 - An ordered graph is a pair G=(G,≺)G=(G,≺) where GG is a graph and ≺ is a total ordering of its vertices. The ordered Ramsey number R̄(G) is the minimum number NN such that every ordered complete graph with NN vertices and with edges colored by two colors contains a monochromatic copy of GG. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings MnMn on nn vertices for which R̄ (Mn) is superpolynomial in nn. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number R̄ (G) is polynomial in the number of vertices of GG if the bandwidth of GG is constant or if GG is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by Károlyi, Pach, Tóth, and Valtr.

AB - An ordered graph is a pair G=(G,≺)G=(G,≺) where GG is a graph and ≺ is a total ordering of its vertices. The ordered Ramsey number R̄(G) is the minimum number NN such that every ordered complete graph with NN vertices and with edges colored by two colors contains a monochromatic copy of GG. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings MnMn on nn vertices for which R̄ (Mn) is superpolynomial in nn. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number R̄ (G) is polynomial in the number of vertices of GG if the bandwidth of GG is constant or if GG is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by Károlyi, Pach, Tóth, and Valtr.

UR - http://www.scopus.com/inward/record.url?scp=85078845229&partnerID=8YFLogxK

U2 - 10.37236/7816

DO - 10.37236/7816

M3 - Article

AN - SCOPUS:85078845229

SN - 1077-8926

VL - 27

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

IS - 1

M1 - P1.16

ER -