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Complete Graph
Edge Coloring
ramsey number
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Canonical Pattern Ramsey Numbers
Canonical Pattern Ramsey Numbers,10.1007/s0037300506036,Graphs and Combinatorics,Maria Axenovich,Robert E. Jamison
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Canonical Pattern Ramsey Numbers
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Citations: 3
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Maria Axenovich
,
Robert E. Jamison
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. 145160, 2005
DOI:
10.1007/s0037300506036
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Citation Context
(2)
...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 Fujita
,
et 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 Axenovich
,
et al.
Edgecolorings avoiding rainbow and monochromatic subgraphs
References
(9)
A Combinatorial Theorem
(
Citations: 63
)
P. Erdos
,
R. Rado
Journal:
Journal of The London Mathematical Societysecond Series  J LONDON MATH SOCSECOND SER
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All Ramsey numbers for cycles in graphs
(
Citations: 47
)
R Faudree
Journal:
Discrete Mathematics  DM
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Ramsey Theory
(
Citations: 396
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Ronald L. Graham
,
Joel H. Spencer
Journal:
Scientific American  SCI AMER
, vol. 263, no. 1, pp. 112117, 1990
On Pattern Ramsey Numbers of Graphs
(
Citations: 12
)
Robert E. Jamison
,
Douglas B. West
Journal:
Graphs and Combinatorics
, vol. 20, no. 3, pp. 333339, 2004
New Upper Bounds for a Canonical Ramsey Problem
(
Citations: 8
)
Tao Jiang
,
Dhruv Mubayi
Journal:
Combinatorica
, vol. 20, no. 1, pp. 141146, 2000
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Citations
(3)
Rainbow Generalizations of Ramsey Theory: A Survey
(
Citations: 2
)
Shinya Fujita
,
Colton Magnant
,
Kenta Ozeki
Journal:
Graphs and Combinatorics
, vol. 26, no. 1, pp. 130, 2010
A note on monotonicity of mixed Ramsey numbers
Maria Axenovich
,
JiHyeok Choi
Published in 2010.
Edgecolorings avoiding rainbow and monochromatic subgraphs
(
Citations: 6
)
Maria Axenovich
,
Perry Iverson
Journal:
Discrete Mathematics  DM
, vol. 308, no. 20, pp. 47104723, 2008