Around P-small subsets of groups
by
Ievgen Lutsenko
Department of Cybernetics, Kyiv University, Ukraine

• P-small if there exists an injective sequence (g_n)_{n ∈ w} in G such that the subsets (g_nA)_{n ∈ w} are pairwise disjoint;

• almost P-small if there exists an injective sequence (g_n)_{n ∈ w} in G such that g_iA∩g_jA is finite for all distinct i, j ∈ w;

• weakly P-small if, for every n ∈ w, there exists elements g₀, ..., g_n of G such that the subsets g₀A, ..., g_nA are pairwise disjoint.

• k-sparse for k ∈ N if G is infinite and, for every infinite subset X of G there exists a non-empty finite subset F of X such that |F| ≤ k and ∩_{g ∈ F}gA is finite;

Theorem 1 Every almost P-small subset can be partitioned in two P-small subsets.

Theorem 2 For every countable group G, there exists an almost P-small subset which is not a weakly P-small.

Theorem 3 For every k ∈ N there exists f(k) ∈ N such that some f(k)-sparse subset of a group G cannot be partitioned in k P-small subsets.

Question 1 Can every k-sparse (or sparse) subset be finitely partitioned to P-small subsets?

Question 2 Does there exist h(k) ∈ N such that every k-sparse subset can be partitioned in h(k) P-small subsets?

Question 3 Can every weakly P-small subset be finitely partitioned to P-small subsets?

Date received: June 27, 2008