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Canonical Pattern Ramsey Numbers

Canonical Pattern Ramsey Numbers,10.1007/s00373-005-0603-6,Graphs and Combinatorics,Maria Axenovich,Robert E. Jamison

Canonical Pattern Ramsey Numbers   (Citations: 3)
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A color pattern is a graph whose edges have been partitioned into color classes. A family of color patterns is a Ramsey family provided there is some sufficiently large integer N such that in any edge coloring of the complete graph KN there is an (isomorphic) copy of at least one of the patterns from . The smallest such N is the Ramsey number of the family . The classical Canonical Ramsey theorem of Erdos and Rado asserts that the family of color patterns is a Ramsey family if it consists of monochromatic, rainbow (totally multicolored) and lexically colored complete graphs. In this paper we treat the asymmetric case by studying the Ramsey number of families containing a rainbow triangle, a lexically colored complete graph and a fixed arbitrary monochromatic graph. In particular we give asymptotically tight bounds for the Ramsey number of a family consisting of rainbow and monochromatic triangle and a lexically colored KN. Among others, we prove some canonical Ramsey results for cycles.
Journal: Graphs and Combinatorics , vol. 21, no. 2, pp. 145-160, 2005
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    • ...Concerning the more general problem of considering general patterns of colors, Jamison and West [74] considered a particular family of colorings (equipartitioned stars) while Axenovich and Jamison [15] studied another family (F = {K lex n , K rain , H mono }). Notice this is related to the rainbow triangle free work dis-...
    • ...Theorem 65 [15] For any connected graph H and any n, there is a constant c = c(n) such that f (n, H ) ≤ cR n−1(H ) (the classical n − 1 color Ramsey number for H)...

    Shinya Fujitaet al. Rainbow Generalizations of Ramsey Theory: A Survey

    • ...Proof. Part a) of the lemma is easy and has been proved in [3], as well as the fact that if (Vi; Vj) is a mixed pair then (Vi; Vl) is not a mixed pair for any l 6= j, which immediately implies part d). We prove part b) by induction on k, which trivially holds for k = 2. Assume that sets V1; V2; : : : ; Vk 1 are ordered so that conclusion of part b) holds...

    Maria Axenovichet al. Edge-colorings avoiding rainbow and monochromatic subgraphs

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