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 topologica...

we introduce some new concepts of topological spaces which say α − separable topologicalspace and o-topological group, α − first axiom, α − second axiom, and we find some relations betweenthem with some applications in normed spaces.

in this paper, we introduce and study the notion of precompacttopological generalized groups and some new results are given.

We consider Class-Uniformly Resolvable Group Divisible Designs (CURGDD), which are resolvable group divisible designs in which each of the resolution classes has the same number of blocks of each size. We derive the fully general necessary conditions including a number of extremal bounds. We present some general constructions including a novel construction for shrinking the index of a master de...

In 1987, Hartman showed that the necessary condition v ≡ 4 or 8 (mod 12) for the existence of a resolvable SQS(v) is also sufficient for all values of v, with 23 possible exceptions. These last 23 undecided orders were removed by Ji and Zhu in 2005 by introducing the concept of resolvable H-designs. In this paper, we first develop a simple but powerful construction for resolvable H-designs, i.e...

We prove that the necessary divisibility conditions are sufficient for the existence of resolvable group divisible designs with a fixed number of sufficiently large groups. Our method combines an application of the Rees product construction with a streamlined recursion based on incomplete transversal designs. With similar techniques, we also obtain new results on decompositions of complete mult...