Ramsey goodness of bounded degree trees
Balla, I. and Pokrovskiy, Alexey and Sudakov, B. (2018) Ramsey goodness of bounded degree trees. Combinatorics, Probability and Computing 27 (3), pp. 289309. ISSN 09635483.

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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 redblue coloring 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 color class in a χ(H)coloring of H. A graph G is called Hgood 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(Hlog4H) then every nvertex bounded degree tree T is Hgood. 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 nvertex bounded degree trees are Hgood when when n≥Ω(H4).
Metadata
Item Type:  Article 

School:  School of Business, Economics & Informatics > Economics, Mathematics and Statistics 
Depositing User:  Alexey Pokrovskiy 
Date Deposited:  21 Jan 2019 10:30 
Last Modified:  12 Jun 2021 17:24 
URI:  https://eprints.bbk.ac.uk/id/eprint/25888 
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