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Ramsey Goodness of Bounded Degree Trees

Part of: Graph theory

Published online by Cambridge University Press:  16 January 2018

IGOR BALLA ,
ALEXEY POKROVSKIY  and
BENNY SUDAKOV
IGOR BALLA
Affiliation:
Department of Mathematics, ETH, 8092 Zurich, Switzerland (e-mail: igor.balla@math.ethz.ch, dr.alexey.pokrovskiy@gmail.com, benjamin.sudakov@math.ethz.ch)
ALEXEY POKROVSKIY
Affiliation:
Department of Mathematics, ETH, 8092 Zurich, Switzerland (e-mail: igor.balla@math.ethz.ch, dr.alexey.pokrovskiy@gmail.com, benjamin.sudakov@math.ethz.ch)
BENNY SUDAKOV
Affiliation:
Department of Mathematics, ETH, 8092 Zurich, Switzerland (e-mail: igor.balla@math.ethz.ch, dr.alexey.pokrovskiy@gmail.com, benjamin.sudakov@math.ethz.ch)
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Abstract

Given a pair of graphs G and H, the Ramsey number R(G, H) is the smallest N such that every red–blue colouring of the edges of the complete graph KN contains a red copy of G or a blue copy of H. If a graph G is connected, it is well known and easy to show that R(G, H) ≥ (|G|−1)(χ(H)−1)+σ(H), where χ(H) is the chromatic number of H and σ(H) is the size of the smallest colour class in a χ(H)-colouring of H. A graph G is called H-good if R(G, H) = (|G|−1)(χ(H)−1)+σ(H). The notion of Ramsey goodness was introduced by Burr and Erdős in 1983 and has been extensively studied since then.

In this paper we show that if n≥ Ω(|H| log4 |H|) then every n-vertex bounded degree tree T is H-good. The dependency between n and |H| is tight up to log factors. This substantially improves a result of Erdős, Faudree, Rousseau, and Schelp from 1985, who proved that n-vertex bounded degree trees are H-good when n ≥ Ω(|H|4).

MSC classification

Primary: 05C55: Generalized Ramsey theory
Secondary: 05C05: Trees
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 3 , May 2018 , pp. 289 - 309
Copyright
Copyright © Cambridge University Press 2018 

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