TY - CHAP

T1 - On Off-Diagonal Ordered Ramsey Numbers of Nested Matchings

AU - Balko, Martin

AU - Poljak, Marian

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.

PY - 2021/1/1

Y1 - 2021/1/1

N2 - For two ordered graphs G< and H<, the ordered Ramsey number r<(G<, H<) is the minimum N such that every red-blue coloring of the edges of the ordered complete graph KN< contains a red copy of G< or a blue copy of H<. For n∈ N, a nested matching NMn< is the ordered graph on 2n vertices with edges { i, 2 n- i+ 1 } for every i= 1, ⋯, n. We improve bounds on the numbers r<(NMn<,K3<) obtained by Rohatgi, we disprove his conjecture about these numbers, and we determine them exactly for n= 4, 5. This gives a stronger lower bound on the maximum chromatic number of k-queue graphs for every k≥ 3. We expand the classical notion of Ramsey goodness to the ordered case and we attempt to characterize all connected ordered graphs that are n-good for every n∈ N. In particular, we discover a new class of such ordered trees, extending all previously known examples.

AB - For two ordered graphs G< and H<, the ordered Ramsey number r<(G<, H<) is the minimum N such that every red-blue coloring of the edges of the ordered complete graph KN< contains a red copy of G< or a blue copy of H<. For n∈ N, a nested matching NMn< is the ordered graph on 2n vertices with edges { i, 2 n- i+ 1 } for every i= 1, ⋯, n. We improve bounds on the numbers r<(NMn<,K3<) obtained by Rohatgi, we disprove his conjecture about these numbers, and we determine them exactly for n= 4, 5. This gives a stronger lower bound on the maximum chromatic number of k-queue graphs for every k≥ 3. We expand the classical notion of Ramsey goodness to the ordered case and we attempt to characterize all connected ordered graphs that are n-good for every n∈ N. In particular, we discover a new class of such ordered trees, extending all previously known examples.

KW - Nested matching

KW - Ordered Ramsey number

KW - Ramsey goodness

UR - http://www.scopus.com/inward/record.url?scp=85114116093&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-83823-2_38

DO - 10.1007/978-3-030-83823-2_38

M3 - Chapter

AN - SCOPUS:85114116093

T3 - Trends in Mathematics

SP - 241

EP - 247

BT - Trends in Mathematics

PB - Springer Science and Business Media Deutschland GmbH

ER -