TY - JOUR

T1 - Ramsey Goodness of Bounded Degree Trees

AU - Balla, Igor

AU - Pokrovskiy, Alexey

AU - Sudakov, Benny

N1 - Publisher Copyright:
© Copyright Cambridge University Press 2018.

PY - 2018/5/1

Y1 - 2018/5/1

N2 - 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 Erdos 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 Erdos, Faudree, Rousseau, and Schelp from 1985, who proved that n-vertex bounded degree trees are H-good when n ≥ Ω(|H|4).

AB - 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 Erdos 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 Erdos, Faudree, Rousseau, and Schelp from 1985, who proved that n-vertex bounded degree trees are H-good when n ≥ Ω(|H|4).

UR - http://www.scopus.com/inward/record.url?scp=85041556418&partnerID=8YFLogxK

U2 - 10.1017/S0963548317000554

DO - 10.1017/S0963548317000554

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AN - SCOPUS:85041556418

SN - 0963-5483

VL - 27

SP - 289

EP - 309

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

IS - 3

ER -