Applications of Ramsey theory in topological dynamics a 1 Aleksandra Kwiatkowska 2 Dana Bartoˇ sov´ Jordi Lopez Abad 3 Brice R. Mbombo 4 1 , 4 University of S˜ ao Paulo 2 UCLA 3 ICMAT Madrid and University of S˜ ao Paulo Forcing and its Applications April 1 The first author was supported by the grants FAPESP 2013/14458-9 and FAPESP 2014/12405-8. Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Outline (KPT) Topological dynamics and Ramsey theory Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Outline (KPT) Topological dynamics and Ramsey theory ( G ) Gurarij space group of linear isometries approximate Ramsey propety for finite dimensional normed spaces Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Outline (KPT) Topological dynamics and Ramsey theory ( G ) Gurarij space group of linear isometries approximate Ramsey propety for finite dimensional normed spaces (S) Poulsen simplex new characterization group of affine homeomorphisms approximate Ramsey propety for finite dimensional simplexes Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Outline (KPT) Topological dynamics and Ramsey theory ( G ) Gurarij space group of linear isometries approximate Ramsey propety for finite dimensional normed spaces (S) Poulsen simplex new characterization group of affine homeomorphisms approximate Ramsey propety for finite dimensional simplexes (L) Lelek fan group of homeomorphism exact Ramsey propety for sequences in FIN k Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Topological dynamics G -flow � X - a continuous action G × X Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Topological dynamics G -flow � X - a continuous action G × X ↑ ↑ topological compact group Hausdorff space Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Topological dynamics G -flow � X - a continuous action G × X ↑ ↑ topological compact group Hausdorff space ex = x g ( hx ) = ( gh ) x Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Topological dynamics G -flow � X - a continuous action G × X ↑ ↑ topological compact group Hausdorff space ex = x g ( hx ) = ( gh ) x X is a minimal G -flow ← → X has no proper closed invariant subset. Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Topological dynamics G -flow � X - a continuous action G × X ↑ ↑ topological compact group Hausdorff space ex = x g ( hx ) = ( gh ) x X is a minimal G -flow ← → X has no proper closed invariant subset. The universal minimal flow M ( G ) is a minimal flow which has every other minimal flow as its factor. Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Topological dynamics G -flow � X - a continuous action G × X ↑ ↑ topological compact group Hausdorff space ex = x g ( hx ) = ( gh ) x X is a minimal G -flow ← → X has no proper closed invariant subset. The universal minimal flow M ( G ) is a minimal flow which has every other minimal flow as its factor. G is extremely amenable ← → its universal minimal flow is a singleton ( ← → every G -flow has a fixed point). Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Structural Ramsey property Theorem (Ramsey) For every k ≤ m and r ≥ 2 , there exists n such that for every colouring of k -element subsets of n with r -many colours there is a subset X of n of size m such that all k -element subsets of X have the same colour. Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Structural Ramsey property Theorem (Ramsey) For every k ≤ m and r ≥ 2 , there exists n such that for every colouring of k -element subsets of n with r -many colours there is a subset X of n of size m such that all k -element subsets of X have the same colour. A class K of finite structures satisfies the Ramsey property if for every A ≤ B ∈ K Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Structural Ramsey property Theorem (Ramsey) For every k ≤ m and r ≥ 2 , there exists n such that for every colouring of k -element subsets of n with r -many colours there is a subset X of n of size m such that all k -element subsets of X have the same colour. A class K of finite structures satisfies the Ramsey property if for every A ≤ B ∈ K and r ≥ 2 a natural number Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Structural Ramsey property Theorem (Ramsey) For every k ≤ m and r ≥ 2 , there exists n such that for every colouring of k -element subsets of n with r -many colours there is a subset X of n of size m such that all k -element subsets of X have the same colour. A class K of finite structures satisfies the Ramsey property if for every A ≤ B ∈ K and r ≥ 2 a natural number there exists C ∈ K such that Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Structural Ramsey property Theorem (Ramsey) For every k ≤ m and r ≥ 2 , there exists n such that for every colouring of k -element subsets of n with r -many colours there is a subset X of n of size m such that all k -element subsets of X have the same colour. A class K of finite structures satisfies the Ramsey property if for every A ≤ B ∈ K and r ≥ 2 a natural number there exists C ∈ K such that for every colouring of copies of A in C by r colours, there is a copy B ′ of B in C, such that all copies of A in B ′ have the same colour. Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Ramsey classes and extremely amenable groups Ramsey classes finite linear orders (Ramsey) finite linearly ordered graphs (Neˇ setˇ ril and R¨ odl) finite linearly ordered metric spaces (Neˇ setˇ ril) finite Boolean algebras (Graham and Rothschild) Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Ramsey classes and extremely amenable groups Ramsey classes finite linear orders (Ramsey) finite linearly ordered graphs (Neˇ setˇ ril and R¨ odl) finite linearly ordered metric spaces (Neˇ setˇ ril) finite Boolean algebras (Graham and Rothschild) Extremely amenable groups Aut( Q , < ) (Pestov) Aut( OR ) – OR the random ordered graph (Kechris, Pestov & Todorˇ cevi´ c) Iso( U , d ) (Pestov) Homeo( C, C ) – ( C, C ) the Cantor space with a generic maximal chain of closed subsets (KPT; Glasner & Weiss) Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
What allows us to use the Ramsey property? A - a first order structures Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
What allows us to use the Ramsey property? A - a first order structures A is ultrahomogeneous ← → every partial finite isomorphism can be extended to an automorphism of A . Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
What allows us to use the Ramsey property? A - a first order structures A is ultrahomogeneous ← → every partial finite isomorphism can be extended to an automorphism of A . G = Aut( A ) with topology of pointwise convergence Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
What allows us to use the Ramsey property? A - a first order structures A is ultrahomogeneous ← → every partial finite isomorphism can be extended to an automorphism of A . G = Aut( A ) with topology of pointwise convergence A - a finitely-generated substructure of A Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
What allows us to use the Ramsey property? A - a first order structures A is ultrahomogeneous ← → every partial finite isomorphism can be extended to an automorphism of A . G = Aut( A ) with topology of pointwise convergence A - a finitely-generated substructure of A G A = { g ∈ G : ga = a ∀ a ∈ A } Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
What allows us to use the Ramsey property? A - a first order structures A is ultrahomogeneous ← → every partial finite isomorphism can be extended to an automorphism of A . G = Aut( A ) with topology of pointwise convergence A - a finitely-generated substructure of A G A = { g ∈ G : ga = a ∀ a ∈ A } form a basis of neighbourhoods of the identity. Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
What allows us to use the Ramsey property? A - a first order structures A is ultrahomogeneous ← → every partial finite isomorphism can be extended to an automorphism of A . G = Aut( A ) with topology of pointwise convergence A - a finitely-generated substructure of A G A = { g ∈ G : ga = a ∀ a ∈ A } form a basis of neighbourhoods of the identity. Theorem (KPT; NvT) Aut( A ) is extremely amenable ← → finitely-generated substructures of A satisfy the Ramsey property and are rigid. Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Universal minimal flows G = Aut( A ) – A ultrahomogeneous Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Universal minimal flows G = Aut( A ) – A ultrahomogeneous G ∗ = Aut( A ∗ ) – A ∗ ultrahomogeneous expansion of A Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
Universal minimal flows G = Aut( A ) – A ultrahomogeneous G ∗ = Aut( A ∗ ) – A ∗ ultrahomogeneous expansion of A Finite substructures of A ∗ satisfy the Ramsey property and are rigid. Dana Bartoˇ sov´ a Ramsey theory in topological dynamics
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