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Published in:
2006 | OriginalPaper | Chapter
On the Induced Ramsey Number IR(P 3, H)
Authors : Alexandr Kostochka, Naeem Sheikh
Published in: Topics in Discrete Mathematics
Publisher: Springer Berlin Heidelberg
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The induced Ramsey number
IR(G, H)
is the least positive integer
N
such that there exists an N-vertex graph
F
with the property that for each 2-coloring of its edges with red and blue, either some induced in
F
subgraph isomorphic to
G
has all its edges colored red, or some induced in
F
subgraph isomorphic to
H
has all its edges colored blue. In this paper, we study
IR
(
P
3
,
H
) for various
H
, where
P
3
is the path with 3 vertices. In particular, we answer a question by Gorgol and Luczak by constructing a family {
H
n
n
=1
∞
such that
$$ \mathop {lim}\limits_{n \to \infty } \sup \tfrac{{IR(P_3 ,H_n )}} {{IR_w (P_3 ,H_n )}} > 1 $$
, where
IR
w
(
G, H
) is defined almost as
IR(G,H)
, with the only difference that
G
should be induced only
in the red subgraph
of
F
(not in
F
itself) and
H
should be induced only in the blue subgraph of
F
.