An extension of the Erdős–Ginzburg–Ziv Theorem to hypergraphs
An n - set partition of a sequence S is a collection of n nonempty subsequences of S , pairwise disjoint as sequences, such that every term of S belongs to exactly one of the subsequences, and the terms in each subsequence are all distinct with the result that they can be considered as sets. For a s...
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| Vydané v: | European journal of combinatorics Ročník 26; číslo 8; s. 1154 - 1176 |
|---|---|
| Hlavný autor: | |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Elsevier Ltd
01.11.2005
|
| ISSN: | 0195-6698, 1095-9971 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | An
n
-
set partition of a sequence
S
is a collection of
n
nonempty subsequences of
S
, pairwise disjoint as sequences, such that every term of
S
belongs to exactly one of the subsequences, and the terms in each subsequence are all distinct with the result that they can be considered as sets. For a sequence
S
, subsequence
S
′
, and set
T
,
|
T
∩
S
|
denotes the number of terms
x
of
S
with
x
∈
T
, and
|
S
|
denotes the length of
S
, and
S
∖
S
′
denotes the subsequence of
S
obtained by deleting all terms in
S
′
. We first prove the following two additive number theory results.
(1) Let
S
be a finite sequence of elements from an abelian group
G
. If
S
has an
n
-set partition,
A
=
A
1
,
…
,
A
n
, such that
|
∑
i
=
1
n
A
i
|
≥
∑
i
=
1
n
|
A
i
|
−
n
+
1
,
then there exists a subsequence
S
′
of
S
, with length
|
S
′
|
≤
max
{
|
S
|
−
n
+
1
,
2
n
}
, and with an
n
-set partition,
A
′
=
A
1
′
,
…
,
A
n
′
, such that
|
∑
i
=
1
n
A
i
′
|
≥
∑
i
=
1
n
|
A
i
|
−
n
+
1
. Furthermore, if
|
|
A
i
|
−
|
A
j
|
|
≤
1
for all
i
and
j
, or if
|
A
i
|
≥
3
for all
i
, then
A
i
′
⊆
A
i
.
(2) Let
S
be a sequence of elements from a finite abelian group
G
of order
m
, and suppose there exist
a
,
b
∈
G
such that
|
(
G
∖
{
a
,
b
}
)
∩
S
|
≤
⌊
m
2
⌋
. If
|
S
|
≥
2
m
−
1
, then there exists an
m
-term zero-sum subsequence
S
′
of
S
with
|
{
a
}
∩
S
′
|
≥
⌊
m
2
⌋
or
|
{
b
}
∩
S
′
|
≥
⌊
m
2
⌋
.
Let
H
be a connected, finite
m
-uniform hypergraph, and
let
f
(
H
)
(
let
f
z
s
(
H
)
)
be the least integer
n
such that for every 2-coloring (coloring with the elements of the cyclic group
Z
m
) of the vertices of the complete
m
-uniform hypergraph
K
n
m
, there exists a subhypergraph
K
isomorphic to
H
such that every edge in
K
is monochromatic (such that for every edge
e
in
K
the sum of the colors on
e
is zero). As a corollary to the above theorems, we show that if every subhypergraph
H
′
of
H
contains an edge with at least half of its vertices monovalent in
H
′
, or if
H
consists of two intersecting edges, then
f
z
s
(
H
)
=
f
(
H
)
. This extends the Erdős–Ginzburg–Ziv Theorem, which is the case when
H
is a single edge. |
|---|---|
| ISSN: | 0195-6698 1095-9971 |
| DOI: | 10.1016/j.ejc.2004.07.005 |