Centerpole sets for colorings of abelian groups

Abstract

A subset C ⊂ G of a group G is called k-centerpole if for each k-coloring of G there is an infinite monochromatic subset G, which is symmetric with respect to a point c ∈ C in the sense that

Symmetry and colorings: Some results and open problems

“…The following theorem generalizes a result of [16] and can be proved by analogy with the proof of Lemmas 13-15 of [10]. Observe that GCH, the Generalized Continuum Hypothesis, is equivalent to the assumption κ = 2 <κ for every infinite cardinal κ.…”

“…Abelian groups G with ν(G) = 2 were characterized in [13]. In [10] the invariant ν(G) was calculated for any Abelian group G. It can be expressed via known group invariants such as the free rank r 0 (G), the cardinality of the group G, and the cardinality of its subgroup B(G) = {g ∈ G : 2g = 0} consisting of elements of order 2.…”

Symmetry and colorings: some results and open problems, II

“…We shall say that a subset C ⊂ R n is r-centerpole if the family of central symmetries S C = {s c : c ∈ C} is (r, 0)-centerpole, which means that ms r (R n , asdim, S C ) ≥ 0. The following theorem proved in [3] and [8] describes the geometry of r-centerpole sets for small r.…”

“…2 r <s = (x i ) ∈ 2 r : r i=1 x i < s and 2 r >s = (x i ) ∈ 2 r : r i=1 x i > s called the s-slices of the r-cube 2 r . The subset Ξ r s = {−1} × 2 r <s ∪ {0} × 2 r <r ∪ {1} × 2 r >s of Z × Z r is called the r s -sandwich.In[8] it is proved that the r s -sandwich is a (r + 1)-centerpole set in Z r+1 for every s ≤ r − 2. The minimal cardinality mins≤r−2 |Ξ r s | = 2 r+1 − 1 − max s≤r−2 r sof such sandwiches yields the upper bound from Theorem 10.3(1).…”