Next week we will have logic seminar.
When: Dec. 30, 10h30
Where: building 72 classroom 123
Speaker: Nadav Meir (BGU)
Our first talk will be an introductory talk about the notions of symmetric indivisibility and elementary indivisibility. We will present the definitions and survey some classic symmetrically indivisible and elementariliy indivisible structures, as proved in [GeKo] and [HKO11].
We will show our new methods of constructing new symmetrically(elementarily) indivisible structures, given known ones. Using these methods we will show an answer to Question 1.
A countable structure M is indivisible if for every colouring of |M| in two colours, there is a monochromatic M ‘ ⊂ M such that M ‘ is isomorphic to M.
The notion of indivisibility is a relatively well-studied notion in combinatorics.
In [GeKo], the notions of symmetric indivisibility was introduced:
Definition: An L-embedding e:M →N is symmetric, if for every σ ∈ aut(M), there is a τ ∈ aut(N) such that e\circ σ = τ\circ e, or in other words, any automorphism of e(M) extends to an automorphism of N.
We say that M is a symmetric substructure of N if the inclusion map, ι:M → N, is symmetric.
Definition: A structure M is symmetrically indivisible if for every colouring of M in two colours, there is a monochromatic M ‘ ⊂ M such that M ‘ is isomorphic to M and M ‘ is a symmetric substructure of \iM .
In [HKO11], the notions of elementary indivisibility was introduced:
Definition: A structure M is elementarily indivisible if for every colouring of M in two colours, there is a monochromatic M ‘ ⊂ M such that M ‘ is isomorphic to M and M ‘ is an elementary substructue of M .
In these papers, several classic known structures were proved to be symmetrically indivisible.
In [HKO11] several open questions have been asked:
Question 1: Let M be a symmetrically indivisible structure in a languageL. Let L0 ⊂ L. Is M \| L0
[GeKo] Stefan Geschke and Menachem Kojman, Symmetrized induced Ramsey theorems. Available at http://www.math.bgu.ac.il/~hassonas/papers/final.pdf
[HKO11] Assaf Hasson, Menachem Kojman and Alf Onshuus. On symmetric indivisibility of countable structures. Available at http://www.math.bgu.ac.il/~kojman/SymPart.pdf