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The indecomposable tournaments T with | W 5
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We consider a tournament
T
=
(
V
,
A
)
. For
X
⊆
V
, the subtournament of
T
induced by
X
is
T
[
X
]
=
(
X
,
A
∩
(
X
×
X
)
)
. An interval of
T
is a subset
X
of
V
such that, for
a
,
b
∈
X
and
x
∈
V
∖
X
,
(
a
,
x
)
∈
A
if and only if
(
b
,
x
)
∈
A
. The trivial intervals of
T
are ∅,
{
x
}
(
x
∈
V
)
and
V
. A tournament is indecomposable if all its intervals are trivial. For
n
⩾
2
,
W
2
n
+
1
denotes the unique indecomposable tournament defined on
{
0
,
…
,
2
n
}
such that
W
2
n
+
1
[
{
0
,
…
,
2
n
−
1
}
]
is the usual total order. Given an indecomposable tournament
T
,
W
5
(
T
)
denotes the set of
v
∈
V
such that there is
W
⊆
V
satisfying
v
∈
W
and
T
[
W
]
is isomorphic to
W
5
. Latka [6] characterized the indecomposable tournaments
T
such that
W
5
(
T
)
=
∅
. The authors [1] proved that if
W
5
(
T
)
≠
∅
, then
|
W
5
(
T
)
|
⩾
|
V
|
−
2
. In this note, we characterize the indecomposable tournaments
T
such that
|
W
5
(
T
)
|
=
|
V
|
−
2
.
MathDoc/Centre Mersenne
Title: The indecomposable tournaments T with
|
W
5
Description:
We consider a tournament
T
=
(
V
,
A
)
.
For
X
⊆
V
, the subtournament of
T
induced by
X
is
T
[
X
]
=
(
X
,
A
∩
(
X
×
X
)
)
.
An interval of
T
is a subset
X
of
V
such that, for
a
,
b
∈
X
and
x
∈
V
∖
X
,
(
a
,
x
)
∈
A
if and only if
(
b
,
x
)
∈
A
.
The trivial intervals of
T
are ∅,
{
x
}
(
x
∈
V
)
and
V
.
A tournament is indecomposable if all its intervals are trivial.
For
n
⩾
2
,
W
2
n
+
1
denotes the unique indecomposable tournament defined on
{
0
,
…
,
2
n
}
such that
W
2
n
+
1
[
{
0
,
…
,
2
n
−
1
}
]
is the usual total order.
Given an indecomposable tournament
T
,
W
5
(
T
)
denotes the set of
v
∈
V
such that there is
W
⊆
V
satisfying
v
∈
W
and
T
[
W
]
is isomorphic to
W
5
.
Latka [6] characterized the indecomposable tournaments
T
such that
W
5
(
T
)
=
∅
.
The authors [1] proved that if
W
5
(
T
)
≠
∅
, then
|
W
5
(
T
)
|
⩾
|
V
|
−
2
.
In this note, we characterize the indecomposable tournaments
T
such that
|
W
5
(
T
)
|
=
|
V
|
−
2
.
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