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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

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Results on set mappings

Results on set mappings

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

15th International Workshop on Set Theory, Luminy, 26 September 2019

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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Results on set mappings

Andr´ as Hajnal (05/13/1931–07/30/2016)

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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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¨

  • tv¨
  • s U. Budapest

Results on set mappings

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Results on set mappings Definition

Paul Tur´ an asked in 1934, if f : R → [R]<ω does there exist an infinite free set.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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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¨

  • tv¨
  • s U. Budapest

Results on set mappings

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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¨

  • tv¨
  • s U. Budapest

Results on set mappings

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Results on set mappings Definition

If κ<κ = κ, let Rκ be the set of all nonconstant f : κ → {0, 1} with no last 0. Order Rκ lexicographically, then we have the notions of noweher dense, everywhere dense, etc.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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SLIDE 8

Results on set mappings Definition

  • Theorem. (Bagemihl) (GCH) If f is a set mapping
  • n Rκ with f (x) nwd for x ∈ Rκ, then there is a

free set of cardinality κ.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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SLIDE 9

Results on set mappings Definition

  • Theorem. (K, with a little help from S.) (κ<κ = κ)

If f is a set mapping on Rκ with f (x) nwd for x ∈ Rκ, then there is an everywhere dense free set.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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Results on set mappings Definition

A set mapping is F : [κ]r → [κ]<µ for some finite r, infinite cardinals κ and µ. A set A ⊆ κ is free if y / ∈ F(u) for u ∈ [A]r, y ∈ A − u.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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SLIDE 11

Results on set mappings Definition

  • Theorem. (Erd˝
  • s–Hajnal) If

F : [expr−1(κ)+]r → [expr−1(κ)+]<κ is a set mapping, then there is a free set of cardinality κ+.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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SLIDE 12

Results on set mappings Finite free sets

  • Theorem. (Kuratowski) If F : [ωn]n → [ωn]<ω is a

set mapping, then there is a free set of size n + 1.

  • Theorem. (Sierpi´

nski) There is a set mapping F : [ωn−1]n → [ωn−1]<ω with no free set of size n + 1.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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Results on set mappings Finite free sets

  • Theorem. (Hajnal–M´

at´ e) If f : [ω2]2 → [ω2]<ω, then there are arbitrarily large finite free sets.

  • Theorem. (Hajnal) If f : [ω3]3 → [ω3]<ω, then

there are arbitrarily large finite free sets.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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Results on set mappings Finite free sets

t0 = 5, t1 = 7, tn+1 is the least number that tn+1 → (tn, 7)5.

  • Theorem. (Komj´

ath–Shelah) It is consistent that there is a set mapping f : [ωn]4 → [ωn]<ω with no free set of cardinality tn.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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SLIDE 15

Results on set mappings Finite free sets

sn is the minimum number such that sn → (5)3

3n.

Roughly a triple exponential.

  • Theorem. (S. Mohsenipour, S. Shelah) It is

consistent that there is a set mapping F : [ωn]4 → [ωn]ω with no free set of size sn.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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SLIDE 16

Results on set mappings Finite free sets

  • Theorem. (Gillibert) If F : [ωn]n → [ωn]<ω is a set

mapping, then there is a free set of size n + 2.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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Results on set mappings Finite free sets

  • Theorem. (Gillibert–Wehrung) If

F : [ωn]r → [ωn]<ω is a set mapping, then there is a free set of size 2⌊ 1

2(1− 1 2r )− n+1 2r ⌋.

For r = 4, this is about 21.016n.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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Results on set mappings Location of image

  • Theorem. (Hajnal–M´

at´ e) Let F : [ω2]2 → [ω2]<ω be a set mapping (a) if β < f (α, β) (α < β < ω2), then there is a free set of size ℵ2; (b) if f (α, β) ⊆ (α, β) (α < β < ω2), then there is an infinite free set.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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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 (µ0, µ1, . . . , µr), if the following holds. For every s ∈ [λ]r with increasing enumeration s = {α0, . . . , αr−1} we have |f (s) ∩ α0| < µ0, |f (s) ∩ (αi, αi+1)| < µi+1 (i < r − 1), and |f (s) ∩ (αr−1, λ)| < µr.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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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

  • rder (κ, κ+, κ++, . . . , κ+r). Then there is a free set
  • f order type κ+ + r − 1.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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Results on set mappings Location of image

  • Theorem. If 1 ≤ r < ω and κ is infinite, then there

is a set mapping fr : [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¨

  • tv¨
  • s U. Budapest

Results on set mappings

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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) ωr−3 + 1, (3 ≤ r < ω).

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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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¨

  • tv¨
  • s U. Budapest

Results on set mappings

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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¨

  • tv¨
  • s U. Budapest

Results on set mappings

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Results on set mappings Free arithmetic progressions

  • Proof. Let V be a vector space over Q, and

B = {bi : i ∈ I} a basis with I ordered. If x ∈ V write as x = λ1bi1 + · · · + λnbin where i1 < · · · < in. Let λ1, . . . λn be the color of x.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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Results on set mappings Free arithmetic progressions

Assume that x, y, z get the same color λ1, . . . , λn and x + z = 2y. Then x = λ1bix

1 + · · · + λnbix n , where ix

1 < · · · < ix n ,

y = λ1biy

1 + · · · + λnbiy n , where iy

1 < · · · < iy n ,

z = λ1biz

1 + · · · + λnbiz n, where iz

1 < · · · < iz n.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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Results on set mappings Free arithmetic progressions

Let i = min{ix

1 , iy 1 , y z 1 }. Then the coefficients of

x, y, z in bi are 0 or λ1, one of them is λ1 and they form a 3-AP. This is only possible, if all are equal to λ1 and so ix

1 = iy 1 = iz 1.

Proceed to ix

2 , iy 2 , iz 2, etc. Eventually, x = y = z.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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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¨

  • tv¨
  • s U. Budapest

Results on set mappings

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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¨

  • tv¨
  • s U. Budapest

Results on set mappings

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Results on set mappings Free arithmetic progressions

  • Theorem. If V is a vector space over Q, |V | = ℵn,

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¨

  • tv¨
  • s U. Budapest

Results on set mappings

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Results on set mappings Free arithmetic progressions

  • Theorem. If V is a vector space over Q with

|V | = ℵn−1, f : V → [V ]<ω is a set mapping, then there is a free arithmetic progression of length n.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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Results on set mappings Free arithmetic progressions

Some old Erd˝

  • s–Hajnal problems

Problem 1. Is the following consistent? GCH plus if f : [ω2]3 → ω2 then there is an uncountable free set. Problem 2. Is the following consistent? GCH plus if f : [ω3]3 → [ω3]<ω then there is an uncountable free set.

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings

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Results on set mappings Free arithmetic progressions

Thank you for your patience!

P´ eter Komj´ ath E¨

  • tv¨
  • s U. Budapest

Results on set mappings