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    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. 289-309. ISSN 0963-5483.

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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 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 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 when n≥Ω(|H|4).

    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: 10 Feb 2021 19:49
    URI: https://eprints.bbk.ac.uk/id/eprint/25888

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