Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Monochrome symmetric subsets in 2-colorings of groups | Monochrome symmetric subsets in 2-colorings of groups Yuliya Gryshko Faculty of Cybernetics Kyiv Taras Shevchenko University 2 korp. 6 03680 Kyiv Ukraine http grishko grishko@ Submitted Jan 31 2002 Accepted Jul 11 2003 Published Aug 3 2003 MR Subject Classifications 05D10 20B07 Abstract A subset A of a group G is called symmetric with respect to the element g E 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
