Yuliya Zelenyuk

Monochrome Symmetric Subsets in Colorings of Finite Abelian Groups

A subset S of a group G is symmetric if there is an element g є G such that gS-1g = S. We study some Ramsey type functions for symmetric subsets in finite Abelian groups.

Transformations and Colorings of Groups

Let G be a compact topological group and let f : G → G be a continuous transformation of G. Define f �: G → G by f �(x) = f(x 1)x and let µ = µG be Haar measure on G. Assume that H = Im fis a subgroup of G and for every measurable C ⊆ H, µG(( f �) 1(C)) = µH(C). Then for every measurable C ⊆ G, there exist S ⊆ C and g ∈ G such that f(Sg 1 ) ⊆ Cg 1 and µ(S) ≥ (µ(C)) 2 .

COUNTING SYMMETRIC BRACELETS

An \(r\)-ary necklace (bracelet) of length \(n\) is an equivalence class of \(r\)-colourings of vertices of a regular \(n\)-gon, taking all rotations (rotations and reflections) as equivalent. A necklace (bracelet) is symmetric if a corresponding colouring is invariant under some reflection. We show that the number of symmetric \(r\)-ary necklaces (bracelets) of length \(n\) is \(\frac{1}{2}(r+1)r^\frac{n}{2}\) if \(n\) is even, and \(r^\frac{n+1}{2}\) if \(n\) is odd. DOI: 10.1017/S0004972713000701

THE NUMBER OF SYMMETRIC COLORINGS OF THE DIHEDRAL GROUP D-3

Abstract We compute the number of symmetric r-colorings and the number of equivalence classes of symmetric r-colorings of the dihedral group D3.

The Number of Symmetric Colorings of the Quaternion Group

We compute the number of symmetric r-colorings and the number of equivalence classes of symmetric r-colorings of the quaternion group.

Ramsey functions for symmetric subsets in compact Abelian groups

Abstract Given a compact group G and r ∈ N, let sr(G) denote the least upper bound of real ϵ > 0 such that for every measurable r-coloring of G, there exists a monochrome symmetric subset of measure ≥ ϵ. A subset A ⊆ G is symmetric if there exists g ∈ G such that gA −1 g = A. We give a general picture of asymptotic behaviour of the function sr(G) for compact Abelian groups.

Symmetric colorings of the dihedral group

ABSTRACT Let G be a finite group and let r∈ℕ. An r-coloring of G is any mapping χ:G→{1,…,r}. Colorings χ and ψ are equivalent if there exists g∈G such that χ(xg−1) = ψ(x) for every x∈G. A coloring χ is symmetric if there exists g∈G such that χ(gx−1g) = χ(x) for every x∈G. Let Sr(G) denote the number of symmetric r-colorings of G and sr(G) the number of equivalence classes of symmetric r-colorings of G. We count Sr(G) and sr(G) in the case where G is the dihedral group Dn.

COUNTING SYMMETRIC COLOURINGS OF THE VERTICES OF A REGULAR POLYGON

A colouring of the vertices of a regular polygon is symmetric if it is invariant under some reflection of the polygon. We count the number of symmetric r-colourings of the vertices of a regular n-gon. DOI: 10.1017/S0004972713001147