Induced graph Ramsey theory

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Induced graph Ramsey theory (2003)

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by Marcus Schaefer , Pradyut Shah

Venue:	Ars Combin
Citations:	3 - 1 self

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@ARTICLE{Schaefer03inducedgraph,
    author = {Marcus Schaefer and Pradyut Shah},
    title = {Induced graph Ramsey theory},
    journal = {Ars Combin},
    year = {2003},
    pages = {3--21}
}

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Abstract

We say that a graph F strongly arrows (G, H) and write F (G, H) if for every edge-coloring of F with colors red and blue a red G or a blue H occurs as an induced subgraph of F. Induced Ramsey numbers are defined by r ∗ (G, H) = min{|V (F) | : F (G, H)}. The value of r ∗ (G, H) is finite for all graphs, and good upper bounds on induced Ramsey numbers in general, and for particular families of graphs are known. Most of these results, however, use the probabilistic method, and therefore do no yield explicit constructions. This paper provides several constructions for upper bounds on r ∗ (G, H) including r ∗ (Cn) = r ∗ (log n)2 (Cn, Cn) ≤ c, r ∗ (T, Kn) ≤ |T |n |T | log |T |, r ∗ (B, Cn) ≤ |B | ⌈log n⌉+4, where T is a tree, B is bipartite, Kn is the complete graph on n vertices and Cn a cycle on n vertices. We also have some new upper bounds for small graphs: r ∗ (K3 + e) ≤ 21, and r ∗ (K4 − e) ≤ 46. 1

Keyphrases

induced graph ramsey theory induced ramsey number probabilistic method particular family good upper bound yield explicit construction several construction induced subgraph complete graph small graph upper bound new upper bound

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