Partitions of groups into sparse subsets
A subset A of a group G is called sparse if, for every infinite subset X of G, there exists a finite subset F ⊂ X, such that ∩x∈FxA is finite. We denote by η(G) the minimal cardinal such that G can be partitioned in η(G) sparse subsets. If |G| > (κ+)א0 then η(G) > κ, if |G| ≤ κ+ then η(G) ≤ κ....
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| Date: | 2012 |
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Інститут прикладної математики і механіки НАН України
2012
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| Cite this: | Partitions of groups into sparse subsets / I. Protasov // Algebra and Discrete Mathematics. — 2012. — Vol. 13, № 1. — С. 107–110. — Бібліогр.: 7 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859678204686499840 |
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| author | Protasov, I. |
| author_facet | Protasov, I. |
| citation_txt | Partitions of groups into sparse subsets / I. Protasov // Algebra and Discrete Mathematics. — 2012. — Vol. 13, № 1. — С. 107–110. — Бібліогр.: 7 назв. — англ. |
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| container_title | Algebra and Discrete Mathematics |
| description | A subset A of a group G is called sparse if, for every infinite subset X of G, there exists a finite subset F ⊂ X, such that ∩x∈FxA is finite. We denote by η(G) the minimal cardinal such that G can be partitioned in η(G) sparse subsets. If |G| > (κ+)א0 then η(G) > κ, if |G| ≤ κ+ then η(G) ≤ κ. We show also that cov(A) ≥ cf|G| for each sparse subset A of an infinite group G, where cov(A) = min{|X| : G = X A}.
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| first_indexed | 2025-11-30T16:38:26Z |
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h.Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 13 (2012). Number 1. pp. 107 – 110
c© Journal “Algebra and Discrete Mathematics”
Partitions of groups into sparse subsets
Igor Protasov
Abstract. A subset A of a group G is called sparse if, for
every infinite subset X of G, there exists a finite subset F ⊂ X, such
that
⋂
x∈F
xA is finite. We denote by η(G) the minimal cardinal such
that G can be partitioned in η(G) sparse subsets. If |G| > (κ+)ℵ0
then η(G) > κ, if |G| 6 κ+ then η(G) 6 κ. We show also that
cov(A) > cf |G| for each sparse subset A of an infinite group G,
where cov(A) = min{|X| : G = XA}.
A subset A of a group G with the identity e is called
• large if there exists a finite subset F such that G = FA;
• small if L \A is large for each large subset L of G;
• thin if gA ∩A is finite for every g ∈ G \ {e};
• sparse if, for every infinite subset X of G, there exists a finite subset
F ⊂ X such that
⋂
x∈F xA is finite.
We note that large, small, and thin subsets can be considered as
asymptotic counterparts of dense, nowhere dense and discrete subsets of
a topological space [7, Chapter 9]. The sparse subsets were introduced
in [2] to characterize the strongly prime ultrafilters in the Stone-Čech
compactification βG of G. If G is infinite then each thin subset is sparse,
and each sparse subset is small [3].
2010 Mathematics Subject Classification: 03E75, 20F99, 20K99.
Key words and phrases: partition of a group, sparse subset of a group.
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108 Partitions of groups into sparse subsets
By [4], every infinite group G can be partitioned in ℵ0 large subsets,
and if G is amenable then G can not be partitioned in > ℵ0 large subsets.
By [5], every infinite group G can be partitioned in ℵ0 small subsets.
For a group G, we denote by µ(G) and η(G) respectively the minimal
cardinals such that G can be partitioned in µ(G) thin subsets and in η(G)
sparse subsets. By [6], µ(G) = |G| if |G| is a limit cardinal, and µ(G) = κ
if G is infinite and |G| = κ+, where κ+ is the cardinal-successor of κ. In
Theorem 1, we evaluate the cardinal η(G).
A covering number of a subset A of G is the cardinal cov(A) =
min{|X| : G = XA}. In Theorem 2, we show that cov(A) > cf |G| for
every sparse subset A of an infinite group G, where cf |G| is the cofinality
of |G|.
Lemma 1. A subset S of a group G is not sparse if and only if there
exists an infinite subset X of G such that, for each infinite subset F of X,
the set {x ∈ G : F−1x ⊆ A} is infinite.
Proof. It suffices to note that x ∈
⋂
g∈F gS if and only if F−1x ⊆ A.
We say that a subset S of a group G is rectangle free if XY * S for
any infinite subsets X,Y of G.
Lemma 2. Every sparse subset S of a group G is rectangle free.
Proof. Apply Lemma 1.
Lemma 3. Let X and Y be infinite sets of cardinality |X| = κ+ and
|Y | > (κ+)λ for some non-zero cardinal λ 6 κ+. For any κ-coloring
χ : X × Y → κ, there are subsets A ⊆ X and Z ⊆ Y such that |A| = λ,
|Z| > (κ+)λ and the set A× Z is monochrome.
Proof. [1, Lemma 1].
Theorem 1. Let G be an infinite group, κ be an infinite cardinal. If
|G| > (κ+)ℵ0 then η(G) > κ. If |G| 6 κ+ then η(G) 6 κ.
Proof. Suppose that |G| > (κ+)ℵ0 , take an arbitrary partition of G into
κ subsets and denote by χ′ corresponding κ-coloring. Then we choose
a subset X of G with |X| = κ+, put Y = G and define a coloring
χ : X ×Y → κ by the rule χ((x, y)) = χ′((x, y)). Applying Lemma 3 with
λ = ℵ0, we get A ⊆ X and Z ⊆ Y such that |A| = ℵ0, |Z| > (κ+)ℵ0 and
A× Z is monochrome. By Lemma 2, A× Z is not sparse, so at least one
cell of the partition is not sparse and η(G) > κ.
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I . Protasov 109
If |G| 6 κ+, by [6, Lemma 2], G can be partitioned in 6 κ thin subsets.
Since every thin subset is sparse, η(G) 6 κ.
Corollary 1. If |G| > 2κ then η(G) > κ.
Proof. It suffices to note that, for any infinite cardinal κ, (κ+)ℵ0 6 (2κ)ℵ0 =
2κ.
Corollary 2. If η(G) = ℵ0 then ℵ0 6 |G| 6 2ℵ0 .
Question 1. Does |G| = 2ℵ0 imply η(G) = ℵ0?
Under CH, Theorem 1 gives an affirmative answer to this question.
To answer this question negatively under ¬CH, it suffices to show that,
for any ℵ0-coloring of ℵ2 × ℵ2, there is a monochrome subset A × B,
A ⊂ ℵ2, B ⊂ ℵ2, |A| = |B| = ℵ0.
Theorem 2. For every sparse subset A of an infinite group G, cov(A) >
cf |G|.
Proof. We suppose the contrary and choose X ⊂ G such that G = XA and
|X| < cf |G|. Clearly, |A| = |G|. Since |X| < cf |A| and A =
⋃
x∈X(A∩xA),
there is x0 ∈ X such that |A ∩ x0A| = |G|. We put A0 = A ∩ x0A so
x0A0 ⊆ A. Suppose that we have chosen distinct elements x0, x1, . . . , xn
of X and the subsets A0 ⊇ A1 ⊇ . . . ⊇ An of A such that |A0| = |A1| =
. . . = |An| = |G| and x0A0 ⊆ A1, x1A1 ⊆ A2, . . . , xnAn ⊆ An+1. We take
an arbitrary element g ∈ G such that g−1X ∩ {x0, . . . , xn} = ∅. Since
|gAn| = |G|, gAn ⊆
⋃
x∈X xA and |X| < cf |G|, there is x ∈ X such that
|gAn ∩ xA| = |G|. We put xn+1 = g−1X, An+1 = An ∩ g−1XA. Then
xn+1 /∈ {x0, x1, . . . , xn}, xn+1An+1 ⊆ An.
After ω steps we get a countable set X ′ = {xn : n ∈ ω} and an
increasing chain {An : n ∈ ω} of subsets of cardinality |G| such that
An+1 ⊆ {g ∈ G : {x0, x1, . . . , xn}g ⊆ A}. By Lemma 1, A is not sparse.
Question 2. Is cov(A) = |G| for every sparse subset A of an arbitrary
infinite group G? By Theorem 2, this is so if |G| is regular.
References
[1] T. Banakh, I. Protasov, Partition of groups and matroids into independent subsets,
Algebra and Discrete Math, 10(2010), 1–7.
[2] M. Filali, Ie. Lutsenko, I.V. Protasov, Boolean group ideals and the ideal structure
of βG, Math. Stud., 31(2009), 19–28.
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110 Partitions of groups into sparse subsets
[3] Ie. Lutsenko, I.V. Protasov, Sparse, thin and other subsets of groups, Intern. J.
Algebra Computation, 19(2009), 491–510.
[4] I.V. Protasov, Partition of groups into large subsets, Math. Notes, 73 (2003), 271–
281.
[5] I.V. Protasov, Small systems of generators of groups, Math. Notes, 76 (2004),
420–426.
[6] I. Protasov, Partition of groups into thin subsets, Algebra and Discrete Math,
11(2011), 88–92.
[7] I. Protasov, M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser. 12,
VNTL, Lviv, 2007.
Contact information
I. Protasov Department of Cybernetics, Kyiv National Uni-
versity, Volodimirska 64, 01033, Kyiv, Ukraine
E-Mail: i.v.protasov@gmail.com
I. Protasov
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| id | nasplib_isofts_kiev_ua-123456789-152190 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-11-30T16:38:26Z |
| publishDate | 2012 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Protasov, I. 2019-06-08T11:08:38Z 2019-06-08T11:08:38Z 2012 Partitions of groups into sparse subsets / I. Protasov // Algebra and Discrete Mathematics. — 2012. — Vol. 13, № 1. — С. 107–110. — Бібліогр.: 7 назв. — англ. 1726-3255 2010 Mathematics Subject Classification: 03E75, 20F99, 20K99. https://nasplib.isofts.kiev.ua/handle/123456789/152190 A subset A of a group G is called sparse if, for every infinite subset X of G, there exists a finite subset F ⊂ X, such that ∩x∈FxA is finite. We denote by η(G) the minimal cardinal such that G can be partitioned in η(G) sparse subsets. If |G| > (κ+)א0 then η(G) > κ, if |G| ≤ κ+ then η(G) ≤ κ. We show also that cov(A) ≥ cf|G| for each sparse subset A of an infinite group G, where cov(A) = min{|X| : G = X A}. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Partitions of groups into sparse subsets Article published earlier |
| spellingShingle | Partitions of groups into sparse subsets Protasov, I. |
| title | Partitions of groups into sparse subsets |
| title_full | Partitions of groups into sparse subsets |
| title_fullStr | Partitions of groups into sparse subsets |
| title_full_unstemmed | Partitions of groups into sparse subsets |
| title_short | Partitions of groups into sparse subsets |
| title_sort | partitions of groups into sparse subsets |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/152190 |
| work_keys_str_mv | AT protasovi partitionsofgroupsintosparsesubsets |