Recent progress in Subset Combinatorics of Groups
We systematize and analyze some results obtained in Subset Combinatorics of G groups after publications the previous surveys [1–4]. The main topics: the dynamical and descriptive characterizations of subsets of a group relatively their combinatorial size, Ramsey-product subsets in connection with so...
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| description | We systematize and analyze some results obtained in Subset Combinatorics of G groups after publications the previous surveys [1–4]. The main topics: the dynamical and descriptive characterizations of subsets of a group relatively their combinatorial size, Ramsey-product subsets in connection with some general concept of recurrence in G-spaces, new ideals in the Boolean algebra PG of all subsets of a group G and in the Stone-Cech compactification βG of G, the combinatorial derivation.
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Український математичний вiсник
Том 14 (2017), № 4, 532 – 547
Recent progress in Subset Combinatorics of
Groups
Igor V. Protasov, Ksenia D. Protasova
Abstract. We systematize and analyze some results obtained in Subset
Combinatorics of G groups after publications the previous surveys [1–4].
The main topics: the dynamical and descriptive characterizations of
subsets of a group relatively their combinatorial size, Ramsey-product
subsets in connection with some general concept of recurrence in G-
spaces, new ideals in the Boolean algebra PG of all subsets of a group
G and in the Stone-Čech compactification βG of G, the combinatorial
derivation.
2010 MSC. 20A05, 20F99, 22A15, 06E15, 06E25.
Key words and phrases. Large, small, thin, thick, sparse and scat-
tered subsets of groups; descriptive complexity; Boolean algebra of sub-
sets of a group; Stone-Čech compactification; ultracompanion; Ramsey-
product subset of a group; recurrence; combinatorial derivation.
1. Introduction
In this paper, we systematize and analyze some results obtained in
Subset Combinatorics of Groups after publications the surveys [1–4].
The main topics: the descriptive and dynamical characterizations of sub-
sets of a group with respect to their combinatorial size, Ramsey-product
subsets in connection with some general concept of recurrence, new ide-
als in the Boolean algebra PG of all subsets of G and in the Stone-Čech
compactification βG of G, the combinatorial derivation.
In these investigations, the principal part play ultrafilters on a group
G. On one hand, ultrafilters are using as a tool to get some purely
combinatorial results. On the other hand, the Subset Combinatorics of
Groups allows to prove new facts about ultrafilters, in particular, about
the Stone-Čech compactification βG of G. In this connection, we recall
some basic definitions concerning ultrafilters.
A filter F on a set X is a family of subsets of X such that
Received 03.12.2017
ISSN 1810 – 3200. c⃝ Iнститут прикладної математики i механiки НАН України
I. V. Protasov, K. D. Protasova 533
• ∅ /∈ F , X ∈ F ;
• A,B ∈ F =⇒ A
∩
B ∈ F ;
• A ∈ F , A ⊆ C =⇒ C ∈ F .
The family of all filters on X is partially ordered by inclusion. A filter
maximal in this ordering is called an ultrafilter. A filter F is an ultrafilter
if and only if X = A
∪
B implies A ∈ F or B ∈ F .
Now we endow X with the discrete topology and identity the Stone-
Čech compactification βX with the set of all ultrafilters on X. An ultra-
filter F is principal if there exists x ∈ X such that F = {A ⊆ X : x ∈ A}.
Otherwise,
∩
F = ∅ and F is called free. Thus, X is identified with the
set of all principal ultrafilters and the set of all free ultrafilter on X is
denoted by X∗.
To describe the topology on βX, given any A ⊆ X we denote Ā =
{F ∈ X : A ∈ F}. Then the set {Ā : A ⊆ X} is a base for the topology
of X. The characteristic topological property of βX: every mapping
f : X −→ K, K is a compact Hausdorff space, can be extended to the
continuous mapping fβ : βX −→ K.
Given a filter φ on X, the set φ̄ = {p ∈ βX : φ ⊆ p} is closed in βX,
and for every non-empty closed subset K of βX, there is a filter φ on X
such that φ̄ = K.
Now let G be a discrete group. Using the characteristic property
of βG, we can extend the group multiplication on G to the semigroup
multiplication on βG in such a way that, for every g ∈ G, the mapping
βG −→ G : p 7−→ gp is continuous and, for every q ∈ βG, the mapping
βG −→ βG : p 7−→ pq is continuous.
To define the product pq of ultrafilters p and q, we take an arbitrary
P ∈ p and, for each x ∈ P , pick some Qx ∈ q. Then,
∪
x∈P xQx is
a member of pq, and each member of pq contains some subsets of this
form.
For properties of the compact right topological semigroup βG and a
plenty of its combinatorial application see [5].
2. Diversity of subsets and ultracompanions
Let G be a group with the identity e, FG denotes the family of all
finite subsets of G. We say that a subset A of G is
• large if G = FA for some F ∈ FG;
• small if L \A is large for every large subset L;
534 Recent progress in Subset Combinatorics of Groups
• extralarge if G \A is small;
• thin if gA ∩A is finite for each g ∈ G \ {e};
• thick if, for every F ∈ FG, there exists a ∈ A such that Fa ⊆ A;
• prethick if FA is thick for some F ∈ FG;
• n-thin, n ∈ N if, for every distinct elements g0, . . . , gn ∈ G, the set
g0A ∩ · · · ∩ gnA is finite;
• sparse if, for every infinite subset X of G, there exists a finite subset
F ⊂ X such that
∩
g∈F gA is finite.
Remark 2.1. In Topological dynamics, large subsets are known as
syndetic, and a subset is small if and only if it fails to be piecewise
syndetic. In [4], the authors use the dynamical terminology.
All above definitions can be unified with usage the following notion [6].
Given a subset A of a group G and an ultrafilter p ∈ G∗, we define a p-
companion of A by
∆p(A) = A∗ ∩Gp = {gp : g ∈ G, A ∈ gp}.
Then, for every infinite group G, the following statement hold:
• A is large if and only if ∆p(A) ̸= ∅ for each p ∈ G∗;
• A is small if and only if, for every p ∈ G∗ and every F ∈ FG, we
have ∆p(FA) ̸= Gp;
• A is thick if and only if, there exist p ∈ G∗ such that ∆p(A) = Gp;
• A is thin if and only if, ∆p(A) ≤ 1 for every p ∈ G∗;
• A is n-thin if and only if, ∆p(A) ≤ n for every p ∈ G∗;
• A is sparse if and only if, ∆p(A) is finite for each p ∈ G∗.
Following [1], we say that a subset A of G is scattered if, for every
infinite subset X of A, there is p ∈ X∗ such that ∆p(X) is finite. Equiv-
alently [7, Theorem 1], A is scattered if each subset ∆p(A) is discrete in
G∗.
Comments. For motivations of above definitions see [1], for more
delicate classification of subsets of a group and G-spaces see [2, 8].
I. V. Protasov, K. D. Protasova 535
3. The descriptive look at the size of subsets of groups
Given a group G, we denote by PG and FG the Boolean algebra of
all subsets of G and its ideal of all finite subsets. We endow PG with the
topology arising from identification (via characteristic functions) of PG
with {0, 1}G. For K ∈ FG the sets
{X ∈ PG : K ⊆ X}, {X ∈ PG : X ∩K = ∅}
form the subbase of this topology.
After the topologization, each family F of subsets of a group G can be
considered as a subspace of PG, so one can ask about the Borel complexity
of F , the question typical in the Descriptive Set Theory (see [9]). We ask
these questions for the most intensively studied families in Combinatorics
of Groups.
For a group G, we denote by LG, ELG, SG, TG, PTG the sets of all
large, extralarge, small, thick and prethick subsets of G, respectively.
Theorem 3.1.For a countable group G, we have: LG is Fσ, TG is
Gδ, PTG is Gδσ, SG and ELG are Fσδ.
A subset A of a group G is called
• P -small if there exists an injective sequence (gn)n∈ω in G such that
the subsets {gnA : n ∈ ω} are pairwise disjoint;
• weakly P -small if, for any n ∈ ω, there exists g0, . . . , gn such that
the subsets g0A, . . . , gnA are pairwise disjoint;
• almost P -small if there exists an injective sequence (gn)n∈ω in G
such that gnA ∩ gmA is finite for all distinct n,m;
• near P -small if, for every n ∈ ω, there exists g0, . . . , gn such that
giA ∩ gjA is finite for all distinct i, j ∈ {0, . . . , n}.
Every infinite group G contains a weakly P -small set, which is not
P -small, see [10]. Each almost P -small subset can be partitioned into
two P -small subsets [8]. Every countable Abelian group contains a near
P -small subset which is neither weakly nor almost P -small [11].
Theorem 3.2.For a countable group G, the sets of thin, weakly P -
small and near P -small subsets of G are Fδσ.
We recall that a topological space X is Polish if X is homeomorphic
to a separable complete metric space. A subset A of a topological space
536 Recent progress in Subset Combinatorics of Groups
X is analytic if A is a continuous image of some Polish space, and A is
coanalytic if X \A is analytic.
Using the classical tree technique [9] adopted to groups in [12], we
get.
Theorem 3.3. For a countable group G, the ideal of sparse subsets
is coanalytic and the set of P -small subsets is analytic in PG.
Given a discrete group G, we identify the Stone-Čech compactifi-
cation βG with the set of all ultrafilters on G and consider βG as a
right-topological semigroup (see Introduction). Each non-empty closed
subspace X of βG is determined by some filter φ on G:
X =
∩
{Φ : Φ ∈ φ}, Φ = {p ∈ βG : Φ ∈ p}.
On the other hand, each filter φ on G is a subspace of PG, so we can
ask about complexity of X as the complexity of φ in PG.
The semigroup βG has the minimal ideal KG which play one of the
key parts in combinatorial applications of βG. By [5], Theorem 1.5, the
closure cl(KG) is determined by the filter of all extralarge subsets of G.
If G is countable, applying Theorem 3.1, we conclude that cl(KG) has
the Borel complexity Fσδ.
An ultrafilter p on G is called strongly prime if p /∈ cl(G∗G∗), where
G∗ is a semigroup of all free ultrafilters on G. We put X = cl(G∗G∗) and
choose the filter φX which determine X. By [13], A ∈ φX if and only
if G\A is sparse. If G is countable, applying Theorem 3.3, we conclude
that φX is coanalitic in PG.
Let (gn)n∈ω be an injective sequence in G. The set
{gi1gi2 . . . gin : 0 ≤ i1 < i2 < . . . < in < ω}
is called an FP-set. By the Hindman Theorem 5.8 [5], for every finite
partition of G, at least one cell of the partition contains an FP -set. We
denote by FPG the family of all subsets of G containing some FP -set.
A subset A of G belongs to FPG if and only if A is an element of some
idempotent of βG. By analogy with Theorem 3.3, we can prove that
FPG is analytic in PG.
Comments. This section reflects the results from [14].
4. The dynamical look at the subsets of a group
Let G be a group. A topological space X is called a G-space if there
is the action X ×G −→ X : (x, g) 7−→ xg such that, for each g ∈ G, the
I. V. Protasov, K. D. Protasova 537
mapping X −→ X : x 7−→ xg is continuous.
Given any x ∈ X and U ⊆ X, we set
[U ]x = {g ∈ G : xg ∈ U}
and denote
O(x) = {xg : g ∈ G}, T (x) = clO(x),
W (x) = {y ∈ T (X) : [U ]x is infinite for each neighbourhood U of y}.
We recall also that x ∈ X is a recurrent point if x ∈W (x).
Now we identify PG with the space {0, 1}G, endow PG with the prod-
uct topology and consider PG as a G-space with the action defined by
A 7−→ Ag, Ag = {ag : a ∈ A}.
We say that a subset A of G is recurrent if A is a recurrent point in
(PG, G).
All groups in this sections are supposed to be infinite.
Theorem 4.1. For a subset A of a group G, the following statements
hold
(i) A is finite if and only if W (A) = ∅;
(ii) A is thick if and only if G ∈W (A).
Theorem 4.2. For a subset A of a group G, the following statements
hold
(i) A is n-thin if and only if |Y | ≤ n for every Y ∈W (A);
(ii) A is sparse if and only if each subset Y ∈W (A) is finite;
(iii) A is scattered if and only if, for every subset B ⊆ A there exists
Y ∈ FG in the closure of {Bbb1 : b ∈ B}.
Let (gn)n∈ω be an injective sequence in G. The set
FP (gn)n∈ω = {gi1gi2 . . . gin : 0 ≤ i1 < i2 < . . . < in < ω}
is called an FP -set.
Given a sequence (bn)n∈ω in G, the set
{gi1gi2 . . . ginbin : 0 ≤ i1 < i2 < . . . < in < ω}
538 Recent progress in Subset Combinatorics of Groups
is called a (right) piecewise shifted FP -set [7].
Theorem 4.3. For a subset A of a group G, the following statements
hold
(i) A is not n-thin if and only if there exist F ∈ [G]n+1 and an
injective sequence (xn)n<ω in G such that Fxn ⊆ A for each n ∈ ω;
(ii) A is not sparse if and only if there exists two injective sequences
(xn)n<ω and (yn)n<ω such that xnym ∈ A for each 0 ≤ n ≤ m < ω;
(iii) A is not scattered if and only if A contains a piecewise shifted
FP -set;
(iv) A contains a recurrent subset if and only if there exists x ∈ A
and an FP -set Y such that xY ⊆ A.
Corollary 4.1. Every scattered subset of a group G has no recurrent
points.
Remark 4.1. By [4, Theorem 2], every scattered subset A of an
amenable group G is absolute null, i.e. µ(A) = 0 for every left invariant
Banach measure µ on G. But this statement could not be generalized
to subsets with no recurrent points. By [17, Theorem 11.6], there is a
subset A of Z of positive Banach measure such that (a+ B) \ A ̸= ∅ for
any FP -set B. By Theorem 4.3(iv), A has no recurrent subsets.
Remark 4.2. Let G be an arbitrary infinite group. In [15], we
constructed two injective sequences (xn)n∈ω, (yn)n∈ω in G such the set
{xnym : 0 ≤ n ≤ m < ω} is scattered. By Theorem 4.3(ii), this subset is
not sparse.
Comments. This section reflects the first part of [15].
5. Ramsey-product subsets and recurrence
In this section, all groups under consideration are supposed to be
infinite; a countable set means a countably infinite set.
Let G be a group and let −→m = (m1 . . . ,mk) ∈ Zk be a number vector
of length k ∈ N. We say that a subset A of a group G is a Ramsey −→m-
product subset if every infinite subset X of G contains pairwise distinct
elements x1, . . . , xk ∈ X such that,
xm1
σ(1) x
m2
σ(2) . . . x
mk
σ(k) ∈ A
I. V. Protasov, K. D. Protasova 539
for every substitution σ ∈ Sk.
Theorem 5.1. For a group G and a number vector −→m=(m1,. . .,mk)∈
Zk, the following statements hold:
(i) a subset A of G is a Ramsey −→m-product subset if and only if
every infinite subset X of G contains a countable subset Y such that
ym1
1 . . . ymk
k ∈ A for any distinct elements y1, . . . , yk ∈ Y .
(ii) the family φ−→m of all Ramsey −→m-product subsets of G is a filter.
For t ∈ Z and q ∈ G∗ we denote by q∧t the ultrafilter with the base
{xt : x ∈ Q}, Q ∈ q. Warning: q∧t and qt are different things. Certainly,
q∧t = qt only if t ∈ {−1, 0, 1}.
We remind the reader that, for a filter φ on G, φ = {p ∈ βG : φ ⊆ p}.
Theorem 5.2. For every group G and any number vector −→m =
(m1, . . . ,mk) ∈ Zk, we have
φ−→m = cl{(q∧m1) . . . (q
∧mk) : q ∈ G∗}.
Now we consider some special cases of vectors m⃗.
Proposition 5.1. For any totally bounded topological group G, any
neighborhood U of the identity e of G is a Ramsey m⃗-product subset for
any vector m⃗ = (m1, . . . ,mk) such that m1 + . . .+mk = 0.
We recall that a quasi-topological group is a group G endowed with a
topology such that, for any a, b ∈ G and ε ∈ 1, 1, the mapping G −→ G :
x 7−→ axεb, is continuous.
Proposition 5.2. The closure Ā of any Ramsey (−1, 1)-product set
A in a quasi-topological group G is a neighborhood of the identity.
Proposition 5.3. Let m⃗ = (m1, . . . ,mk) be a number vector and
s = m1 + . . . +mk. For any Ramsey m⃗-product subset A of a group G,
the set {xs : x ∈ G} is contained in the closure of A in any non-discrete
group topology on G.
Proposition 5.4. Let G be the Boolean group of all finite subsets of
Z, endowed with the group operation of symmetric difference. The set
A = G \ {{x, y} : x, y ∈ Z, 0 ̸= x− y ∈ {z3 : z ∈ Z}}
has the following properties:
540 Recent progress in Subset Combinatorics of Groups
(i) A is a Ramsey m⃗-product for any vector m⃗ = (m1, . . . ,mk) ∈
(2Z+ 1)k of length k ≥ 2;
(ii) A does not contain the difference BB−1 of any large subset B of
G;
(iii) A is not a neighborhood of zero in a totally bounded group topol-
ogy on G.
Now we show how Ramsey (−1, 1)-product sets arise in some general
concept of recurrence on G-spaces.
Let G be a group with the identity e and let X be a G-space with the
action G×X −→ X, (g, x) 7−→ gx. If X = G and gx is the product of g
and x then X is called a left regular G-space.
Given a G-space X, a family F of subset of X and A ∈ F, we denote
∆F(A) = {g ∈ G : gB ⊆ A for some B ∈ F, B ⊆ A}.
Clearly, e ∈ ∆F(A) and if F is upward directed (A ∈ F, A ⊆ C imply
C ∈ F) and if F is G-invariant (A ∈ F, g ∈ G imply gA ∈ F) then
∆F(A) = {g ∈ G : gA ∩A ∈ F},∆F(A) = (∆F(A))
−1.
If X is a left regular G-space and ∅ /∈ F then ∆F(A) ⊆ AA−1.
For a G-space X and a family F of subsets of X, we say that a subset
R of G is F-recurrent if ∆F(A) ∩ R ̸= ∅ for every A ∈ F. We denote by
RF the filter on G with the base ∩{∆F′(A) : A ∈ F′}, where F′ is a finite
subfamily of F, and note that, for an ultrafilter p on G, RF ∈ p if and
only if each member of p is F-recurrent.
The notion of an F-recurrent subset is well-known in the case in which
G is an amenable group, X is a left regular G-space and F = {A ⊆ X :
µ(A) > 0 for some left invariant Banach measure µ on X}. See [16–18]
for historical background.
We recall [19] that a filter φ on a group G is left topological if φ
is a base at the identity e for some (uniquely defined) left translation
invariant (each left shift x 7−→ gx is continuous) topology on G. If φ is
left topological then φ is a subsemigroup of βG [19]. If G = X and a
filter φ is left topological then φ = Rφ.
Proposition 5.5. For every G-space X and any family F of subsets
of X, the filter RF is left topological.
Let X be a G-space and let F be a family of subsets of X. We say
that a family F′ of subsets of X is F-disjoint if A∩B /∈ F for any distinct
A,B ∈ F′.
I. V. Protasov, K. D. Protasova 541
A family F′ of subsets of X is called F-packing large if, for each A ∈ F′,
any F-disjoint family of subsets of X of the form gA, g ∈ G is finite.
Proposition 5.6. Let X be a G-space and let F be a G-invariant
upward directed family of subsets of X. Then F is F-packing large if and
only if, for each A ∈ F, the set △F(A) is a Ramsey (-1,1)-product set.
Applying Theorem 5.2, we conclude that △F(A) contains all ultrafil-
ters of the form q−1q, q ∈ G∗, and in the case X = G, G is amenable and
F is the family of all subsets of positive Banach measure, we get Theorem
3.14 from [18].
Comments. The proofs of all above statements can be find in [20,21].
6. Ideals in PG and βG
We recall that a family I of subsets of a set X is an ideal in the
Boolean algebra PG of all subsets of G if G /∈ I and A ∈ I, B ∈ I,
C ⊆ A imply A ∪ B ∈ I, C ∈ I. A family φ of subsets of G is a filter if
and only if the family {X \A : A ∈ φ} is an ideal.
For an infinite group G, an ideal I in PG is called left (right) trans-
lation invariant if gA ∈ I (Ag ∈ I) for all g ∈ G, A ∈ I. If I is left and
right translation invariant then I is called translation invariant. Clearly,
each left (right) translation invariant ideal of G contains the ideal FG of
all finite subsets of G. An ideal I in PG is called a group ideal if FG ⊆ I
and if A ∈ I, B ∈ I then AB−1 ∈ I.
Now we endow G with the discrete topology and use the standard
extension of the multiplication on G to the semigroup multiplication on
βG, see Introduction.
It follows directly from the definition of the multiplication in βG that
G∗, G∗G∗ are ideals in the semigroup βG, and G∗ is the unique maximal
closed ideal in βG. By Theorem 4.44 from [5], the closure K(βG) of the
minimal ideal K(G) of βG is an ideal, so K(βG) is the smallest closed
ideal in βG. For the structure of K(βG) and some other ideals in βG
see [5, Sections 4, 6].
For an ideal I in PG, we put
I∧ = {p ∈ βG : G \A ∈ p for each A ∈ I},
and use the following observations:
• I is left translation invariant if and only if I∧ is a left ideal of the
semigroup βG ;
542 Recent progress in Subset Combinatorics of Groups
• I is right translation invariant if and only if (I∧)G ⊆ I∧.
We use also the inverse to ∧ mapping ∨. For a closed subset K of βG,
we take the unique filter φ on G such that K = φ and put
K∨ = {G \A : A ∈ φ}.
In this section, all groups under consideration are suppose to be infi-
nite.
We denote by SmG, ScG, SpG the families of all small, scattered
and sparse subsets of a group G. These families are translation invariant
ideals in PG (see [6, Proposition 1]), and for every group G, the following
inclusions are strict [6, Proposition 12]
SpG ⊂ ScG ⊂ SmG.
We say that a subset A of G is finitely thin if A is n-thin for some
n ∈ N. The family FTG of all finitely thin subsets of G is a translation
invariant ideal in PG which contains the ideal < TG > generated by the
family of all thin subsets of G. By [22, Theorem 1.2] and [23, Theorem 3],
if G is either countable or Abelian and |G| < ℵω then FTG =< TG >.
By [23, Example 3], there exists an Abelian group G of cardinality ℵω
such that < TG >⊂ FTG.
Theorem 6.1. For every group G, we have Sm∧
G = K(βG).
This is Theorem 4.40 from [5] in the form given in [24, Theorem 12.5].
Theorem 6.2. For every group G, Sp∧G = G∗G∗.
This is Theorem 10 from [13].
6.1. Between G∗G∗ and G∗.
Theorem 6.3. For every group G, the following statements hold:
(i) if I is a left translation invariant ideal in PG and I ≠ FG then
there exists a left translation invariant ideal J in PG such that FG ⊂
J ⊂ I and J ⊂ SpG;
(ii) if I is a right translation invariant ideal in PG and I ≠ FG then
there exists a right translation invariant J in PG such that FG ⊂ J ⊂ I;
(iii) if G is either countable or Abelian and I is a translation invari-
ant ideal in PG such that I ≠ FG then there exists a translation invariant
ideal J in PG such that FG ⊂ J ⊂ I and J ⊂ SpG.
I. V. Protasov, K. D. Protasova 543
Theorem 6.4. For every group G, the following statements hold:
(i) if L is a closed left ideal in βG such that L ⊂ G∗ then there exists
a closed left ideal L′ of βG such that L ⊂ L′ ⊂ G∗, G∗G∗ ⊂ L′;
(ii) if R is a closed subset of G∗ such that R ̸= G∗ and RG ⊆ R then
there exists a closed subset R′ of G∗ such that R ⊂ R′ ⊂ G∗, R′G ⊆ R;
(iii) if G is either countable or Abelian and I is a closed ideal in
βG such that I ⊂ G∗ then there exists a closed ideal I ′ in βG such that
I ⊂ I ′ ⊂ G∗, G∗G∗ ⊂ I.
For a cardinal κ, Sκ denotes the group of all permutations of κ.
Theorem 6.5. For every infinite cardinal κ, there exists a closed
ideal I in βSκ such that
(i) S∗
κS
∗
κ ⊂ I;
(ii) if M is a closed ideal in βSκ and I ⊆M ⊆ G∗ then either M = I
or M = S∗
κ.
Theorem 6.6. For every group G, we have FTG ⊂ SpG so G∗G∗ ⊂
FT∧
G .
For subsets X,Y of a group G, we say that the product XY is an
n-stripe if |X| = n, n ∈ N and |Y | = ω. It is easy to see that a subset A
of G is n-thin if and only if A has no (n + 1)-stripes. Thus, p ∈ FT∧
G is
and only if each member P ∈ p has an n-stripe for every n ∈ N.
We say thatXY is an (n,m)-rectangle if |X| = n, |Y | = m , n,m ∈ N.
We say that a subset A of G has bounded rectangles if there is n ∈ N such
that A has no (n, n)-rectangles (and so (n,m)-rectangles for each m > n).
We denote by BRG the family of all subsets of G with bounded rect-
angles.
Theorem 6.7. For a group G, the following statements hold:
(i) BRG is a translation invariant ideal in PG;
(ii) BR∧
G is a closed ideal in βG and p ∈ BR∧
G if and only if each
member P ∈ p has an (n, n)-rectangle for every n ∈ N;
(iii) BRG ⊂ FTG.
544 Recent progress in Subset Combinatorics of Groups
6.2. Between K(G) and G∗G∗.
Theorem 6.8. For a group G, the following statements hold:
(i) Sc∧G = cl{ϵp : ϵ ∈ G∗, p ∈ βG, ϵϵ = ϵ};
(ii) Sc∧G is an ideal in βG and p ∈ Sc∧G if and only if each member
of p contains a piecewise shifted FP -set;
(iii) Sc∧G is the minimal closed ideal in βG containing all idempotents
of G∗.
For a group G, we put IG,n = G∗, IG,n+1 = G∗IG,n and note that
IG,n is an ideal in βG.
Theorem 6.9. For every group G and n ∈ ω, we have
(i) IG,n+1 ⊂ IG,n
(ii) Sc∧G ⊂ IG,n.
For a natural number n, we denote by (G∗)n the product of n copies
of n. Clearly, (G∗)n+1 ⊆ (G∗)n. and (G∗)n ⊆ IG,n.
Theorem 6.10. For every group G and n ∈ ω, we have
(i) (G∗)n+1 ⊂ (G∗)n;
(ii) Sc∧G ⊂ (G∗)n.
Comments. This section is an extract from [25].
7. The combinatorial derivation
Let G be a group with the identity e. For a subset A of G, we denote
△(A) = {g ∈ G : |gA
∩
A = ∞|},
observe that (△(A))−1 = △(A), △(A) ⊆ AA−1, and say that the map-
ping
△ : PG −→ PG, A 7−→ △(A)
is the combinatorial derivation.
Theorem 7.1. For an infinite group G and a subset A of G, the
following statements hold
I. V. Protasov, K. D. Protasova 545
(1) A is finite if and only if △(A) = ∅;
(2) △(A) = {e} if and only if A is infinite and thin;
(3) if A is thick then △(A) = G;
(4) if A is prethick then △(A) is large.
Theorem 7.2. Every infinite group G contains a subset A such that
G = AA−1 and △(A) = {e}.
Theorem 7.3. Let A be a subset of an infinite group G such that A =
A−1. Then there exist two thin subsets X, Y of G such that △(X
∪
Y ) =
A.
We consider also the inverse to △, multivalued mapping ∇ defined by
∇(A) = {B ⊆ G : △(B) = A}.
For a family F of subsets of a group G, we say that F is △-complete
(∇-complete) if △(A) ∈ F (∇(A) ⊆ F) for each A ∈ F .
Theorem 7.4. For every infinite group G, the following statements
hold
(1) the families of all small and sparse subsets of G is ∇-complete;
(2) if an ideal I in PG is △-complete and ∇-complete then I = PG;
(3) If I is a group ideal in PG, I ̸= PG, then I is △-complete and I
is contained in the ideal of all small subsets of G.
Comments. More information on combinatorial derivation in [26–28].
In particular, Theorem 6.2 from [26] shows that the trajectory A −→
△(A) −→ △2(A) −→ . . . of a subset A of G could be surprisingly com-
plicated: stabilizing, increasing, decreasing, periodic or chaotic. Also [26]
contains some parallels between the combinatorial and topological deriva-
tions.
References
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[2] I. Protasov, S. Slobodianiuk, On the subset combinatorics of G-spaces // Algebra
Discrete Math., 17 (2011), 98–109.
[3] I. Protasov, S. Slobodianiuk, Partitions of groups // Math. Stud., 42 (2014),
115–128.
[4] T. Banakh, I. Protasov, S. Slobodianiuk, Densities, submeasures and partitions
of groups // Algebra Discrete Math., 17 (2014), 193–221.
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applications, Berlin, New York: Walter de Gruyter, 1998.
546 Recent progress in Subset Combinatorics of Groups
[6] I. Protasov, S. Slobodianiuk, Ultracompanions of subsets of a group // Comment.
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[9] A. Kechris, Classical Descriptive Set Theory, Springer, 1995.
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Intern. J. Algebra Comput., 19 (2008), 1–6.
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[12] T. Banakh, N. Lyaskovska, On thin-complete ideals of subsets of groups // Ukr.
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[14] T. Banakh, I. Protasov, K. Protasova, Descriptive complexity of the sizes of sub-
sets of groups // Ukr. Mat. J., 69 (2017), No. 9, 1280–1283.
[15] I. Protasov, S. Slobodianiuk, The dynamical look at the subsets of a group //
Appl. Gen. Topol., 16 (2015), No. 2, 217–224.
[16] H. Furstenberg, Poincare recurrence and number theory // Bull. Amer. Math.
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[17] N. Hindman, Ultrafilters and combinatorial number theory // Lecture Notes in
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[20] I. Protasov, K. Protasova, On recurrence in G-spaces // Algebra Discrete Math.,
23 (2017), No. 2, 80–85.
[21] T. Banakh, I. Protasov, K. Protasova, Ramsey-product subsets of a group //
Math. Stud., 47 (2017), 145–149.
[22] Ie. Lutsenko, I. Protasov, Thin subsets of balleans // Appl. Gen. Topology, 11
(2010), 89–93.
[23] I. Protasov, S. Slobodianiuk, Thin subsets of groups // Ukr. Math. J., 65 (2013),
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[24] I. Protasov, T. Banakh, Ball Structures and Colorings of Graphs and Groups //
Math. Stud. Monogr. Ser, 11, Lviv: VNTL Publisher, 2003.
[25] I. Protasov, K. Protasova, Ideals in PG and βG // ArXiv: 1704.02494–1.
[26] I. Protasov, The combinatorial derivation // Appl. Gen. Topology, 14 (2013),
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[27] I. Protasov, The combinatorial derivation and its inverse mapping // Central
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[28] J. Erde, A note on combinatorial derivation // arxiv: 1210. 7622.
I. V. Protasov, K. D. Protasova 547
Contact information
Igor V. Protasov Faculty of Computer Science and
Cybernetics of Taras Shevchenko
National University of Kyiv,
Kyiv, Ukraine
E-Mail: i.v.protasov@gmail.com
Ksenia D.
Protasova
Faculty of Computer Science and
Cybernetics of Taras Shevchenko
National University of Kyiv,
Kyiv, Ukraine
E-Mail: ksuha@freenet.com.ua
|
| id | nasplib_isofts_kiev_ua-123456789-169376 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| isbn | 2010 MSC. 20A05, 20F99, 22A15, 06E15, 06E25 |
| issn | 1810-3200 |
| language | English |
| last_indexed | 2025-12-01T13:33:51Z |
| publishDate | 2017 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Protasov, I.V. Protasova, K.D. 2020-06-11T17:53:39Z 2020-06-11T17:53:39Z 2017 Recent progress in Subset Combinatorics of Groups / I.V. Protasov, K.D. Protasova // Український математичний вісник. — 2017. — Т. 14, № 4. — С. 532-547. — Бібліогр.: 28 назв. — англ. 2010 MSC. 20A05, 20F99, 22A15, 06E15, 06E25 1810-3200 https://nasplib.isofts.kiev.ua/handle/123456789/169376 We systematize and analyze some results obtained in Subset Combinatorics of G groups after publications the previous surveys [1–4]. The main topics: the dynamical and descriptive characterizations of subsets of a group relatively their combinatorial size, Ramsey-product subsets in connection with some general concept of recurrence in G-spaces, new ideals in the Boolean algebra PG of all subsets of a group G and in the Stone-Cech compactification βG of G, the combinatorial derivation. en Інститут прикладної математики і механіки НАН України Український математичний вісник Recent progress in Subset Combinatorics of Groups Article published earlier |
| spellingShingle | Recent progress in Subset Combinatorics of Groups Protasov, I.V. Protasova, K.D. |
| title | Recent progress in Subset Combinatorics of Groups |
| title_full | Recent progress in Subset Combinatorics of Groups |
| title_fullStr | Recent progress in Subset Combinatorics of Groups |
| title_full_unstemmed | Recent progress in Subset Combinatorics of Groups |
| title_short | Recent progress in Subset Combinatorics of Groups |
| title_sort | recent progress in subset combinatorics of groups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/169376 |
| work_keys_str_mv | AT protasoviv recentprogressinsubsetcombinatoricsofgroups AT protasovakd recentprogressinsubsetcombinatoricsofgroups |