Monochrome Symmetric Subsets in 2-Colorings of Groups

A subset A of a group G is called symmetric with respect to the element g ∈ G if A = gA−1g. It is proved that in any 2-coloring, every infinite group G contains monochrome symmetric subsets of arbitrarily large cardinality < |G|. A topological space is called resolvable if it can be partitioned into two dense subsets [8]. In [4] W. Comfort and J. van Mill proved that each nondiscrete topological Abelian group with finitely many elements of order 2 is resolvable. In that paper it was also posed the problem of describing of absolutely resolvable groups. A group is called absolutely resolvable if it can be partitioned into two subsets dense in any nondiscrete group topology. This problem turned out to be rather difficult even for rational group Q [11], and for real group R it had remained unsolved. In Abelian case this problem was finally solved by Y. Zelenyuk who proved that each infinite Abelian group with finitely many elements of order 2 is absolutely resolvable [13]. It is easy to see that an Abelian group G is absolutely resolvable if and only if it can be partitioned into two subsets not containing subsets of the form g + U where U is a neighborhood of zero in some nondiscrete group topology. In [10] I. Protasov considered a question close to the above problem. He described Abelian groups which can be partitioned into two subsets not containing infinite subsets of the form g + U where U = −U . Such subsets were called symmetric and groups that can be partitioned into two subsets not containing infinite symmetric subsets – assymetrically resolvable. More precisely, there was given the following equivalent definition of a symmetric subset. A subset A of an Abelian group G is called symmetric with respect to the element g ∈ G if 2g − A = A. Later on R. Grigorchuk extended this definition to arbitrary groups. A subset A of a group G is called symmetric with respect to the element g ∈ G if gA−1g = A. This notion turned out to be enough fruitful, especially against a background of Ramsey Theory (see surveys [1,2]). According [10] an infinite Abelian group is assymetrically resolvable if and only if it is either a direct product of an infinite cyclic group and a finite Abelian group or a countable periodic Abelian group with finitely many elements of order 2. The problem the electronic journal of combinatorics 10 (2003), #R28 1 of describing of all assymetrically resolvable groups is considerably more complicated. For example, it was open for the free group on two generators [2, problem 1.2], and also for each infinite finitely generated periodic group. In case of infinite finitely generated groups of finite torsion, it was not even known whether there exist arbitrarily large finite monochrome symmetric subsets in any 2-coloring [2, problem 1.7]. In this note, the first theorem states if the commutator subgroup G′ of a group G contains a finitely generated subgroup different from an almost cyclic group, then G is not assymetrically resolvable. Recall that an almost cyclic group is a group containing a cyclic subgroup of finite index. In particular, every finite group is almost cyclic. By the first theorem, it follows that both the free group on two generators and every infinite finitely generated periodic group are not assymetrically resolvable. Next, by means of this result we prove more two theorems. One theorem states that in any 2coloring, every infinite group G contains monochrome symmetric subsets of arbitrarily large cardinality < G. Another theorem concerns the problem of describing of all assymetrically resolvable groups. It states that every such group G is either almost cyclic or countable locally finite provided G′ is finite or G′ is infinite and G/G′ is periodic. The proof of the first theorem uses the following nontrivial fact: every group of linear growth is almost cyclic. Indeed, every group of polynomial growth contains a nilpotent subgroup G of finite index [6] and a degree d of a polynomial is evaluated by means of the lower central series G = G1 > G2 > · · · , Gk+1 = [G, Gk],