Results on set mappings Results on set mappings P´ eter Komj´ ath E¨ otv¨ os U. Budapest 15th International Workshop on Set Theory, Luminy, 26 September 2019 P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Andr´ as Hajnal (05/13/1931–07/30/2016) P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Definition A set mapping is F : κ → P ( κ ) for some infinite cardinal κ . A set A ⊆ κ is free if y / ∈ F ( u ) for u ∈ A , y ∈ A − { u } . P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Definition an asked in 1934, if f : R → [ R ] <ω does Paul Tur´ there exist an infinite free set. P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Definition Fundamental Theorem on Set Mappings. (Hajnal) If κ > µ , F : κ → [ κ ] <µ then there is a free set of size κ . P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Definition Theorem. (Bagemihl) If f is a set mapping on R with f ( x ) nowhere dense for x ∈ R then there is an everywhere dense free set. P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Definition If κ <κ = κ , let R κ be the set of all nonconstant f : κ → { 0 , 1 } with no last 0. Order R κ lexicographically, then we have the notions of noweher dense, everywhere dense, etc. P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Definition Theorem. (Bagemihl) (GCH) If f is a set mapping on R κ with f ( x ) nwd for x ∈ R κ , then there is a free set of cardinality κ . P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Definition Theorem. (K, with a little help from S.) ( κ <κ = κ ) If f is a set mapping on R κ with f ( x ) nwd for x ∈ R κ , then there is an everywhere dense free set. P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Definition A set mapping is F : [ κ ] r → [ κ ] <µ for some finite r , infinite cardinals κ and µ . ∈ F ( u ) for u ∈ [ A ] r , A set A ⊆ κ is free if y / y ∈ A − u . P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Definition Theorem. (Erd˝ os–Hajnal) If F : [exp r − 1 ( κ ) + ] r → [exp r − 1 ( κ ) + ] <κ is a set mapping, then there is a free set of cardinality κ + . P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Finite free sets Theorem. (Kuratowski) If F : [ ω n ] n → [ ω n ] <ω is a set mapping, then there is a free set of size n + 1. Theorem. (Sierpi´ nski) There is a set mapping F : [ ω n − 1 ] n → [ ω n − 1 ] <ω with no free set of size n + 1. P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Finite free sets e) If f : [ ω 2 ] 2 → [ ω 2 ] <ω , Theorem. (Hajnal–M´ at´ then there are arbitrarily large finite free sets. Theorem. (Hajnal) If f : [ ω 3 ] 3 → [ ω 3 ] <ω , then there are arbitrarily large finite free sets. P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Finite free sets t 0 = 5, t 1 = 7, t n +1 is the least number that t n +1 → ( t n , 7) 5 . Theorem. (Komj´ ath–Shelah) It is consistent that there is a set mapping f : [ ω n ] 4 → [ ω n ] <ω with no free set of cardinality t n . P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Finite free sets s n is the minimum number such that s n → (5) 3 3 n . Roughly a triple exponential. Theorem. (S. Mohsenipour, S. Shelah) It is consistent that there is a set mapping F : [ ω n ] 4 → [ ω n ] ω with no free set of size s n . P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Finite free sets Theorem. (Gillibert) If F : [ ω n ] n → [ ω n ] <ω is a set mapping, then there is a free set of size n + 2. P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Finite free sets Theorem. (Gillibert–Wehrung) If F : [ ω n ] r → [ ω n ] <ω is a set mapping, then there is a free set of size 2 r ) − n +1 2 r ⌋ . 2 ⌊ 1 2 (1 − 1 For r = 4, this is about 2 1 . 016 n . P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Location of image e) Let F : [ ω 2 ] 2 → [ ω 2 ] <ω Theorem. (Hajnal–M´ at´ be a set mapping (a) if β < f ( α, β ) ( α < β < ω 2 ), then there is a free set of size ℵ 2 ; (b) if f ( α, β ) ⊆ ( α, β ) ( α < β < ω 2 ), then there is an infinite free set. P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Location of image Definition. If λ is an infinite cardinal, 1 ≤ r < ω , we call a set mapping f : [ λ ] r → P ( λ ) of order ( µ 0 , µ 1 , . . . , µ r ), if the following holds. For every s ∈ [ λ ] r with increasing enumeration s = { α 0 , . . . , α r − 1 } we have | f ( s ) ∩ α 0 | < µ 0 , | f ( s ) ∩ ( α i , α i +1 ) | < µ i +1 ( i < r − 1), and | f ( s ) ∩ ( α r − 1 , λ ) | < µ r . P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Location of image Theorem. (GCH) Assume that 0 < r < ω , λ = κ + r . Let f : [ λ ] r → P ( λ ) be a set mapping of order ( κ, κ + , κ ++ , . . . , κ + r ). Then there is a free set of order type κ + + r − 1. P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Location of image Theorem. If 1 ≤ r < ω and κ is infinite, then there is a set mapping f r : [ κ + r ] r → P ( κ + r ) of order (0 , κ + , κ ++ , . . . , κ + r − 1 , 0), with no free set of order type κ + + r . P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Location of image Theorem. If 1 ≤ r < ω , κ is infinite, then there is a set mapping f : [ κ + r ] r → P ( κ + r ) of order ( κ + , 0 , 0 , . . . , 0) such that f has no free set of order type 2 , ( r = 1) ω, ( r = 2) ω r − 3 + 1 , (3 ≤ r < ω ) . P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Free arithmetic progressions Two methods of decomposing vector spaces into the union of countably many parts each omitting some configuration. P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Free arithmetic progressions Theorem. (Rado) Each vector space over Q is the union of ctbly many pieces, each omitting a 3-AP. P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Free arithmetic progressions Proof. Let V be a vector space over Q , and B = { b i : i ∈ I } a basis with I ordered. If x ∈ V write as x = λ 1 b i 1 + · · · + λ n b i n where i 1 < · · · < i n . Let � λ 1 , . . . λ n � be the color of x . P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Free arithmetic progressions Assume that x , y , z get the same color � λ 1 , . . . , λ n � and x + z = 2 y . Then n , where i x 1 < · · · < i x x = λ 1 b i x 1 + · · · + λ n b i x n , n , where i y 1 < · · · < i y y = λ 1 b i y 1 + · · · + λ n b i y n , n , where i z 1 < · · · < i z z = λ 1 b i z 1 + · · · + λ n b i z n . P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Free arithmetic progressions 1 , i y Let i = min { i x 1 , y z 1 } . Then the coefficients of x , y , z in b i are 0 or λ 1 , one of them is λ 1 and they form a 3-AP. This is only possible, if all are equal to 1 = i y λ 1 and so i x 1 = i z 1 . 2 , i y Proceed to i x 2 , i z 2 , etc. Eventually, x = y = z . P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Free arithmetic progressions Definition. If S is a set, H is a set system on S , then the coloring number of H is countable, Col ( H ) ≤ ω , if there is a well ordering < of S such that for each x ∈ S , x is the largest element of finitely many sets in H . P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Free arithmetic progressions If a Rado-type proof gives that for some vector space V and configuration system H on V , V is the union of countably many parts omitting configurations in H , do we have Col ( H ) ≤ ω ? P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
Results on set mappings Free arithmetic progressions Theorem. If V is a vector space over Q , | V | = ℵ n , then there is a well ordering such that each element is the last member of only finitely many arithmetic progressions of length n + 1. Consequently, there is a set mapping f : V → [ V ] <ω with no free arithmetic progression of length n + 1. P´ eter Komj´ ath E¨ otv¨ os U. Budapest Results on set mappings
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