On Some Three-Color Ramsey Numbers for Paths
Abstract
For graphs
G
1
,
G
2
,
G
3
, the three-color Ramsey number
R(
G
1
,
G
2
,
G
3
)
is the smallest integer
n
such that if we arbitrarily color the edges of the complete graph of order
n
with 3 colors, then it contains a monochromatic copy of
G
i
in color
i
, for some
1≤i≤3
.
First, we prove that the conjectured equality
R
3
(
C
2n
,
C
2n
,
C
2n
)=4n
, if true, implies that
R
3
(
P
2n+1
,
P
2n+1
,
P
2n+1
)=4n+1
for all
n≥3
. We also obtain two new exact values
R(
P
8
,
P
8
,
P
8
)=14
and
R(
P
9
,
P
9
,
P
9
)=17
, furthermore we do so without help of computer algorithms. Our results agree with a formula
R(
P
n
,
P
n
,
P
n
)=2n−2+(nmod2)
which was proved for sufficiently large
n
by Gyárfás, Ruszinkó, Sárközy, and Szemerédi in 2007. This provides more evidence for the conjecture that the latter holds for all
n≥1
.