The Open Dihypergraph Dichotomy for Definable Subsets of Generalized Baire Spaces Dorottya Sziráki joint work with Philipp Schlicht Hamburg Set Theory Workshop 2020 The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 1 Dorottya Sziráki
The open graph dichotomy for subsets of κ κ Let κ be an infinite cardinal such that κ <κ = κ . Let X ⊆ κ κ . A graph G on X is an open graph if it is an open subset of X × X . OGD κ ( X ) If G is an open graph on X , then either G has a κ -coloring ( i.e., X is the union of κ many G -independent sets ), or G includes a κ -perfect complete subgraph ( i.e., there is a continuous injection f : κ 2 → X such that ( f ( x ) , f ( y )) ∈ G for all distinct x, y ∈ κ 2 . ) The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 2 Dorottya Sziráki
OGD κ ( X ) for definable subsets X of κ κ Theorem (Feng) 1 OGD ω ( X ) holds for all Σ 1 1 subsets X ⊆ ω ω . The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 3 Dorottya Sziráki
OGD κ ( X ) for definable subsets X of κ κ Theorem (Feng) 1 OGD ω ( X ) holds for all Σ 1 1 subsets X ⊆ ω ω . 2 If λ is inaccessible, then in any Col( ω, <λ ) -generic extension V [ G ] , OGD ω ( X ) holds for all subsets X ⊆ ω ω definable from an element of ω Ord . The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 3 Dorottya Sziráki
OGD κ ( X ) for definable subsets X of κ κ Theorem (Feng) 1 OGD ω ( X ) holds for all Σ 1 1 subsets X ⊆ ω ω . 2 If λ is inaccessible, then in any Col( ω, <λ ) -generic extension V [ G ] , OGD ω ( X ) holds for all subsets X ⊆ ω ω definable from an element of ω Ord . X is definable from an element of ω Ord if X = { x : ϕ ( x, a ) } for some order formula ϕ with a parameter a ∈ ω Ord . The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 3 Dorottya Sziráki
OGD κ ( X ) for definable subsets X of κ κ Theorem (Feng) 1 OGD ω ( X ) holds for all Σ 1 1 subsets X ⊆ ω ω . 2 If λ is inaccessible, then in any Col( ω, <λ ) -generic extension V [ G ] , OGD ω ( X ) holds for all subsets X ⊆ ω ω definable from an element of ω Ord . The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 3 Dorottya Sziráki
OGD κ ( X ) for definable subsets X of κ κ Theorem (Feng) 1 OGD ω ( X ) holds for all Σ 1 1 subsets X ⊆ ω ω . 2 If λ is inaccessible, then in any Col( ω, <λ ) -generic extension V [ G ] , OGD ω ( X ) holds for all subsets X ⊆ ω ω definable from an element of ω Ord . Suppose κ is an uncountable cardinal such that κ <κ = κ . Theorem (Sz.) If λ > κ is inaccessible, then in any Col( κ, <λ ) -generic extension V [ G ] , 1 ( κ ) subsets X ⊆ κ κ . OGD κ ( X ) holds for all Σ 1 The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 3 Dorottya Sziráki
OGD κ ( X ) for definable subsets X of κ κ Theorem (Feng) 1 OGD ω ( X ) holds for all Σ 1 1 subsets X ⊆ ω ω . 2 If λ is inaccessible, then in any Col( ω, <λ ) -generic extension V [ G ] , OGD ω ( X ) holds for all subsets X ⊆ ω ω definable from an element of ω Ord . Suppose κ is an uncountable cardinal such that κ <κ = κ . Theorem (Sz.) If λ > κ is inaccessible, then in any Col( κ, <λ ) -generic extension V [ G ] , 1 ( κ ) subsets X ⊆ κ κ . OGD κ ( X ) holds for all Σ 1 Theorem (Schlicht, Sz.) In Col( κ, <λ ) -generic extensions, where λ > κ is inaccessible, OGD κ ( X ) holds for all subsets X ⊆ κ κ definable from an element of κ Ord . The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 3 Dorottya Sziráki
OGD κ ( X ) for definable subsets X of κ κ Theorem (Feng) 1 OGD ω ( X ) holds for all Σ 1 1 subsets X ⊆ ω ω . 2 If λ is inaccessible, then in any Col( ω, <λ ) -generic extension V [ G ] , OGD ω ( X ) holds for all subsets X ⊆ ω ω definable from an element of ω Ord . Suppose κ is an uncountable cardinal such that κ <κ = κ . Theorem (Sz.) If λ > κ is inaccessible, then in any Col( κ, <λ ) -generic extension V [ G ] , 1 ( κ ) subsets X ⊆ κ κ . OGD κ ( X ) holds for all Σ 1 Theorem (Schlicht, Sz.) In Col( κ, <λ ) -generic extensions, where λ > κ is inaccessible, OGD κ ( X ) holds for all subsets X ⊆ κ κ definable from an element of κ Ord . These results give the exact consistency strength of these statements. The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 3 Dorottya Sziráki
A higher dimensional version Introduced in the κ = ω case by R. Carroy, B. Miller and D. Soukup. The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 4 Dorottya Sziráki
A higher dimensional version Introduced in the κ = ω case by R. Carroy, B. Miller and D. Soukup. Suppose κ <κ = κ ≥ ω . Let X ⊆ κ κ and let 2 ≤ δ ≤ κ . Suppose H is a δ -dimensional dihypergraph on X , i.e., H ⊆ δ X is a set of non- constant sequences. The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 4 Dorottya Sziráki
A higher dimensional version Introduced in the κ = ω case by R. Carroy, B. Miller and D. Soukup. Suppose κ <κ = κ ≥ ω . Let X ⊆ κ κ and let 2 ≤ δ ≤ κ . Suppose H is a δ -dimensional dihypergraph on X , i.e., H ⊆ δ X is a set of non- constant sequences. H is box-open if it is open in the box topology on δ X . The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 4 Dorottya Sziráki
A higher dimensional version Introduced in the κ = ω case by R. Carroy, B. Miller and D. Soukup. Suppose κ <κ = κ ≥ ω . Let X ⊆ κ κ and let 2 ≤ δ ≤ κ . Suppose H is a δ -dimensional dihypergraph on X , i.e., H ⊆ δ X is a set of non- constant sequences. H is box-open if it is open in the box topology on δ X . OGD δ κ ( X, H ) Either H has a κ -coloring (i.e., X is the union of κ many H -independent sets), The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 4 Dorottya Sziráki
A higher dimensional version Introduced in the κ = ω case by R. Carroy, B. Miller and D. Soukup. Suppose κ <κ = κ ≥ ω . Let X ⊆ κ κ and let 2 ≤ δ ≤ κ . Suppose H is a δ -dimensional dihypergraph on X , i.e., H ⊆ δ X is a set of non- constant sequences. H is box-open if it is open in the box topology on δ X . OGD δ κ ( X, H ) Either H has a κ -coloring (i.e., X is the union of κ many H -independent sets), or there is a continuous map f : κ δ → X which is a homomorphism from H δ to H (i.e. f δ ( H δ ) ⊆ H ). The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 4 Dorottya Sziráki
A higher dimensional version Introduced in the κ = ω case by R. Carroy, B. Miller and D. Soukup. Suppose κ <κ = κ ≥ ω . Let X ⊆ κ κ and let 2 ≤ δ ≤ κ . Suppose H is a δ -dimensional dihypergraph on X , i.e., H ⊆ δ X is a set of non- constant sequences. H is box-open if it is open in the box topology on δ X . OGD δ κ ( X, H ) Either H has a κ -coloring (i.e., X is the union of κ many H -independent sets), or there is a continuous map f : κ δ → X which is a homomorphism from H δ to H (i.e. f δ ( H δ ) ⊆ H ). x ∈ δ ( κ δ ) :( ∃ t ∈ <κ δ ) � H δ = ( ∀ α < δ ) t ⌢ � α � ⊂ x α � . OGD δ κ ( X ) OGD δ κ ( X, H ) holds for all δ -dimensional � x 0 , x 1 , . . . , x α , . . . � ∈ H δ box-open dihypergraphs H on X . The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 4 Dorottya Sziráki
OGD δ κ ( X ) For all δ -dimensional box-open dihypergraphs H on X , either H has a κ -coloring, or there is a continuous homomorphism f : κ δ → X from H δ to H . The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 5 Dorottya Sziráki
OGD δ κ ( X ) For all δ -dimensional box-open dihypergraphs H on X , either H has a κ -coloring, or there is a continuous homomorphism f : κ δ → X from H δ to H . Example Let x 0 � = x 1 ∈ κ 2 . Let t be the node where they split. � x 0 , x 1 � ∈ H 2 iff x 0 ( | t | ) = 0 and x 1 ( | t | ) = 1 . The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 5 Dorottya Sziráki
OGD δ κ ( X ) For all δ -dimensional box-open dihypergraphs H on X , either H has a κ -coloring, or there is a continuous homomorphism f : κ δ → X from H δ to H . Example Let x 0 � = x 1 ∈ κ 2 . Let t be the node where they split. � x 0 , x 1 � ∈ H 2 iff x 0 ( | t | ) = 0 and x 1 ( | t | ) = 1 . The smallest graph (i.e. symmetric relation) containing H 2 is the complete graph K 2 on κ 2 . The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 5 Dorottya Sziráki
OGD δ κ ( X ) For all δ -dimensional box-open dihypergraphs H on X , either H has a κ -coloring, or there is a continuous homomorphism f : κ δ → X from H δ to H . Example Let x 0 � = x 1 ∈ κ 2 . Let t be the node where they split. � x 0 , x 1 � ∈ H 2 iff x 0 ( | t | ) = 0 and x 1 ( | t | ) = 1 . The smallest graph (i.e. symmetric relation) containing H 2 is the complete graph K 2 on κ 2 . G is a graph on X . Let f : κ 2 → X be continuous homomorphism from H 2 to G . The Open Dihypergraph Dichotomy for Definable Subsets of κ κ 5 Dorottya Sziráki
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