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Miklós Schweitzer
1984 Miklós Schweitzer
1
1
Part of
1984 Miklós Schweitzer
Problems
(1)
Miklós Schweitzer 1984- Problem 1
Source:
9/4/2016
1. Let
k
k
k
be an arbitrary cardinality. Show that there exists a tournament
T
k
=
(
V
n
,
E
n
)
T_k = (V_n , E_n)
T
k
=
(
V
n
,
E
n
)
such that for any coloring
f
:
E
n
→
k
f: E_n \to k
f
:
E
n
→
k
of the edge set
E
n
E_n
E
n
, there are three different vertices
x
0
,
x
1
,
x
2
∈
V
n
x_0 , x_1 , x_2 \in V_n
x
0
,
x
1
,
x
2
∈
V
n
such that
x
0
x
1
,
x
1
x
2
,
x
2
x
0
∈
E
n
x_0 x_1 , x_1 x_2 , x_2 x_0 \in E_n
x
0
x
1
,
x
1
x
2
,
x
2
x
0
∈
E
n
and
∣
{
f
(
x
0
x
1
)
,
f
(
x
1
x
2
)
,
f
(
x
2
x
0
)
}
∣
≤
2
\left | \{ f(x_0 x_1), f(x_1 x_2), f(x_2 x_0)\} \right |\leq 2
∣
{
f
(
x
0
x
1
)
,
f
(
x
1
x
2
)
,
f
(
x
2
x
0
)}
∣
≤
2
(A tounament is a directed graph such that for any vertices
x
,
y
∈
V
n
,
x
≠
y
x, y \in V_n, x \neq y
x
,
y
∈
V
n
,
x
=
y
exactly one of the relations
x
y
∈
E
n
xy \in E_n
x
y
∈
E
n
holds.) (C.19) [A. Hajnal]
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