On some downwards transfer properties on
Foreman-Laver model
André Ottenbreit Maschio Rodrigues
a joint work with Sakaé Fuchino and Hiroshi Sakai
Kobe University

November 29, 2016

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

1 / 23

Kunen’s Model
Theorem (Kunen)
Assume there exists a huge cardinal κ with target λ. Then there
exists W a forcing extension of V such that κ = (ℵ1 )W , λ = (ℵ2 )W
and there exists an ℵ2 -saturated ℵ1 -ideal in W .
Magidor modified the construction to lessen the hypothesis on κfrom huge to almost huge.
Magidor, Foreman and Shelah actually proved that MM implies
the existence of an ℵ2 -saturated ℵ1 -ideal.
The above model by Kunen satisfies Chang’s conjecture.
By modifying Kunen’s construction, Foreman and Laverconstructed a model with some nice transfer properties from ℵ2
to ℵ1 .
O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

2 / 23

Kunen’s Model
Theorem (Kunen)
Assume there exists a huge cardinal κ with target λ. Then there
exists W a forcing extension of V such that κ = (ℵ1 )W , λ = (ℵ2 )W
and there exists an ℵ2 -saturated ℵ1 -ideal in W .
Magidor modified the construction to lessen the hypothesis on κfrom huge to almost huge.
Magidor, Foreman and Shelah actually proved that MM implies
the existence of an ℵ2 -saturated ℵ1 -ideal.
The above model by Kunen satisfies Chang’s conjecture.
By modifying Kunen’s construction, Foreman and Laverconstructed a model with some nice transfer properties from ℵ2
to ℵ1 .
O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

2 / 23

Kunen’s Model
Theorem (Kunen)
Assume there exists a huge cardinal κ with target λ. Then there
exists W a forcing extension of V such that κ = (ℵ1 )W , λ = (ℵ2 )W
and there exists an ℵ2 -saturated ℵ1 -ideal in W .
Magidor modified the construction to lessen the hypothesis on κfrom huge to almost huge.
Magidor, Foreman and Shelah actually proved that MM implies
the existence of an ℵ2 -saturated ℵ1 -ideal.
The above model by Kunen satisfies Chang’s conjecture.
By modifying Kunen’s construction, Foreman and Laverconstructed a model with some nice transfer properties from ℵ2
to ℵ1 .
O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

2 / 23

Kunen’s Model
Theorem (Kunen)
Assume there exists a huge cardinal κ with target λ. Then there
exists W a forcing extension of V such that κ = (ℵ1 )W , λ = (ℵ2 )W
and there exists an ℵ2 -saturated ℵ1 -ideal in W .
Magidor modified the construction to lessen the hypothesis on κfrom huge to almost huge.
Magidor, Foreman and Shelah actually proved that MM implies
the existence of an ℵ2 -saturated ℵ1 -ideal.
The above model by Kunen satisfies Chang’s conjecture.
By modifying Kunen’s construction, Foreman and Laverconstructed a model with some nice transfer properties from ℵ2
to ℵ1 .

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

2 / 23

Kunen’s Model
Theorem (Kunen)
Assume there exists a huge cardinal κ with target λ. Then there
exists W a forcing extension of V such that κ = (ℵ1 )W , λ = (ℵ2 )W
and there exists an ℵ2 -saturated ℵ1 -ideal in W .
Magidor modified the construction to lessen the hypothesis on κfrom huge to almost huge.
Magidor, Foreman and Shelah actually proved that MM implies
the existence of an ℵ2 -saturated ℵ1 -ideal.
The above model by Kunen satisfies Chang’s conjecture.
By modifying Kunen’s construction, Foreman and Laverconstructed a model with some nice transfer properties from ℵ2
to ℵ1 .

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

2 / 23

Recall:

Definition (Chromatic number)
Given a graph G = hV , E i, a function f : V −→ κ is called aκ-coloring of G if no adjacent vertex have the same value by f . The
chromatic number of G (Chr(G)) is the smallest cardinal κ forwhich there exists a κ-coloring.
We denote:

Definition (Transfer of the chromatic number)
Given cardinals γ < δ, we denote by TrChr (δ, γ) the statement:“ Every graph of size and chromatic number δ has a subgraph of sizeand chromatic number γ. ”

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

3 / 23

Recall:

Definition (Chromatic number)
Given a graph G = hV , E i, a function f : V −→ κ is called aκ-coloring of G if no adjacent vertex have the same value by f . The
chromatic number of G (Chr(G)) is the smallest cardinal κ forwhich there exists a κ-coloring.
We denote:

Definition (Transfer of the chromatic number)
Given cardinals γ < δ, we denote by TrChr (δ, γ) the statement:“ Every graph of size and chromatic number δ has a subgraph of sizeand chromatic number γ. ”

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

3 / 23

The original result

Theorem (Foreman-Laver)Assume there exists a huge cardinal κ with target λ. Then there
exists W a forcing extension of V such that κ = (ℵ1 )W , λ = (ℵ2 )W
and which satisfies TrChr (ℵ2 , ℵ1 ).
Notice that the failure of the same transfer property is consistent:

Theorem (Baumgartner)
Assume CH. Then, there exists a countably closed ℵ2 -c.c. poset B ofsize ℵ2 which forces ¬TrChr (ℵ2 , ℵ1 ).

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

4 / 23

The original result

Theorem (Foreman-Laver)Assume there exists a huge cardinal κ with target λ. Then there
exists W a forcing extension of V such that κ = (ℵ1 )W , λ = (ℵ2 )W
and which satisfies TrChr (ℵ2 , ℵ1 ).
Notice that the failure of the same transfer property is consistent:

Theorem (Baumgartner)
Assume CH. Then, there exists a countably closed ℵ2 -c.c. poset B ofsize ℵ2 which forces ¬TrChr (ℵ2 , ℵ1 ).

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

4 / 23

Construction of the model - 1

Suppose κ is a huge cardinal. Let j : V −→ M be an elementaryembedding with crit(j)=κ, λ := j(κ) and M λ ⊂ M.
the construction consists in building a two step iteration P ∗ Ṙ,and V P∗Ṙ will be the desired model.
P collapses κ to ℵ1 , while R collapses λ to ℵ2 .
P is κ-c.c., while R is λ-c.c and < κ-closed.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

5 / 23

Construction of the model - 1

Suppose κ is a huge cardinal. Let j : V −→ M be an elementaryembedding with crit(j)=κ, λ := j(κ) and M λ ⊂ M.
the construction consists in building a two step iteration P ∗ Ṙ,and V P∗Ṙ will be the desired model.
P collapses κ to ℵ1 , while R collapses λ to ℵ2 .
P is κ-c.c., while R is λ-c.c and < κ-closed.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

5 / 23

Construction of the model - 1

Suppose κ is a huge cardinal. Let j : V −→ M be an elementaryembedding with crit(j)=κ, λ := j(κ) and M λ ⊂ M.
the construction consists in building a two step iteration P ∗ Ṙ,and V P∗Ṙ will be the desired model.
P collapses κ to ℵ1 , while R collapses λ to ℵ2 .
P is κ-c.c., while R is λ-c.c and < κ-closed.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

5 / 23

Construction of the model - 1

Suppose κ is a huge cardinal. Let j : V −→ M be an elementaryembedding with crit(j)=κ, λ := j(κ) and M λ ⊂ M.
the construction consists in building a two step iteration P ∗ Ṙ,and V P∗Ṙ will be the desired model.
P collapses κ to ℵ1 , while R collapses λ to ℵ2 .
P is κ-c.c., while R is λ-c.c and < κ-closed.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

5 / 23

Constructing the model - 2

We fix G ∗ H a P ∗ Ṙ generic over V .
We can find Gb ∗ Hb a j(P) ∗ j(Ṙ) generic over V such thatj : V −→ M can be extended to an elementary embeddingb
J : V [G ∗ H] −→ M[Gb ∗ H].
Gb is chosen so that V [G ∗ H] ⊂ V [Gb].
V [Gb] is a generic extension of V [G ∗ H] by a κ-centred forcing
poset (union of κ-many centred sets of conditions).
b is a < λ-closed generic extension of V [Gb].
V [Gb ∗ H]

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

6 / 23

Constructing the model - 2

We fix G ∗ H a P ∗ Ṙ generic over V .
We can find Gb ∗ Hb a j(P) ∗ j(Ṙ) generic over V such thatj : V −→ M can be extended to an elementary embeddingb
J : V [G ∗ H] −→ M[Gb ∗ H].
Gb is chosen so that V [G ∗ H] ⊂ V [Gb].
V [Gb] is a generic extension of V [G ∗ H] by a κ-centred forcing
poset (union of κ-many centred sets of conditions).
b is a < λ-closed generic extension of V [Gb].
V [Gb ∗ H]

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

6 / 23

Constructing the model - 2

We fix G ∗ H a P ∗ Ṙ generic over V .
We can find Gb ∗ Hb a j(P) ∗ j(Ṙ) generic over V such thatj : V −→ M can be extended to an elementary embeddingb
J : V [G ∗ H] −→ M[Gb ∗ H].
Gb is chosen so that V [G ∗ H] ⊂ V [Gb].
V [Gb] is a generic extension of V [G ∗ H] by a κ-centred forcing
poset (union of κ-many centred sets of conditions).
b is a < λ-closed generic extension of V [Gb].
V [Gb ∗ H]

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

6 / 23

Constructing the model - 2

We fix G ∗ H a P ∗ Ṙ generic over V .
We can find Gb ∗ Hb a j(P) ∗ j(Ṙ) generic over V such thatj : V −→ M can be extended to an elementary embeddingb
J : V [G ∗ H] −→ M[Gb ∗ H].
Gb is chosen so that V [G ∗ H] ⊂ V [Gb].
V [Gb] is a generic extension of V [G ∗ H] by a κ-centred forcing
poset (union of κ-many centred sets of conditions).
b is a < λ-closed generic extension of V [Gb].
V [Gb ∗ H]

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

6 / 23

Constructing the model - 2

We fix G ∗ H a P ∗ Ṙ generic over V .
We can find Gb ∗ Hb a j(P) ∗ j(Ṙ) generic over V such thatj : V −→ M can be extended to an elementary embeddingb
J : V [G ∗ H] −→ M[Gb ∗ H].
Gb is chosen so that V [G ∗ H] ⊂ V [Gb].
V [Gb] is a generic extension of V [G ∗ H] by a κ-centred forcing
poset (union of κ-many centred sets of conditions).
b is a < λ-closed generic extension of V [Gb].
V [Gb ∗ H]

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

6 / 23

Sketch of the proof - 1
We show that V [G ∗ H] |= “ TrChr (ℵ2 , ℵ1 ) ”.
In V [G ∗ H], let G = hλ, E i be a graph of chromatic number
λ = ℵ2 .
We want to show thatV [G ∗ H] |= “ ∃G 0 ⊆ G, |G 0 | = Chr(G 0 ) = ℵ1 = κ ”

By elementarity, it is enough to show that :b |= “ ∃G 0 < J(G), |G 0 | = Chr(G 0 ) = J(κ) = λ = ℵ1 ”
M[Gb ∗ H]
we claim that J 00 G is such a subgraph. We use the assumptionb
M λ ⊂ M to show that indeed J 00 G ∈ M[Gb ∗ H].
O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

7 / 23

Sketch of the proof - 1
We show that V [G ∗ H] |= “ TrChr (ℵ2 , ℵ1 ) ”.
In V [G ∗ H], let G = hλ, E i be a graph of chromatic number
λ = ℵ2 .
We want to show thatV [G ∗ H] |= “ ∃G 0 ⊆ G, |G 0 | = Chr(G 0 ) = ℵ1 = κ ”

By elementarity, it is enough to show that :b |= “ ∃G 0 < J(G), |G 0 | = Chr(G 0 ) = J(κ) = λ = ℵ1 ”
M[Gb ∗ H]
we claim that J 00 G is such a subgraph. We use the assumptionb
M λ ⊂ M to show that indeed J 00 G ∈ M[Gb ∗ H].

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

7 / 23

Sketch of the proof - 1
We show that V [G ∗ H] |= “ TrChr (ℵ2 , ℵ1 ) ”.
In V [G ∗ H], let G = hλ, E i be a graph of chromatic number
λ = ℵ2 .
We want to show thatV [G ∗ H] |= “ ∃G 0 ⊆ G, |G 0 | = Chr(G 0 ) = ℵ1 = κ ”

By elementarity, it is enough to show that :b |= “ ∃G 0 < J(G), |G 0 | = Chr(G 0 ) = J(κ) = λ = ℵ1 ”
M[Gb ∗ H]
we claim that J 00 G is such a subgraph. We use the assumptionb
M λ ⊂ M to show that indeed J 00 G ∈ M[Gb ∗ H].

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

7 / 23

Sketch of the proof - 1
We show that V [G ∗ H] |= “ TrChr (ℵ2 , ℵ1 ) ”.
In V [G ∗ H], let G = hλ, E i be a graph of chromatic number
λ = ℵ2 .
We want to show thatV [G ∗ H] |= “ ∃G 0 ⊆ G, |G 0 | = Chr(G 0 ) = ℵ1 = κ ”

By elementarity, it is enough to show that :b |= “ ∃G 0 < J(G), |G 0 | = Chr(G 0 ) = J(κ) = λ = ℵ1 ”
M[Gb ∗ H]
we claim that J 00 G is such a subgraph. We use the assumptionb
M λ ⊂ M to show that indeed J 00 G ∈ M[Gb ∗ H].

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

7 / 23

Sketch of the proof - 1
We show that V [G ∗ H] |= “ TrChr (ℵ2 , ℵ1 ) ”.
In V [G ∗ H], let G = hλ, E i be a graph of chromatic number
λ = ℵ2 .
We want to show thatV [G ∗ H] |= “ ∃G 0 ⊆ G, |G 0 | = Chr(G 0 ) = ℵ1 = κ ”

By elementarity, it is enough to show that :b |= “ ∃G 0 < J(G), |G 0 | = Chr(G 0 ) = J(κ) = λ = ℵ1 ”
M[Gb ∗ H]
we claim that J 00 G is such a subgraph. We use the assumptionb
M λ ⊂ M to show that indeed J 00 G ∈ M[Gb ∗ H].

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

7 / 23

Sketch of the proof - 2
Suppose, towards a contradiction, that J 00 G has chromaticbnumber ℵ0 in M[Gb ∗ H].
b we have that G has countable
Since J 00 G ≈ G in V [Gb ∗ H],bchromatic number in V [Gb ∗ H].
b is < λ close, so
The forcing poset to get from V [Gb] to V [Gb ∗ H]
we can apply the following lemma:

Lemma (< λ-closed transfer)
Let D a < λ-closed poset. If D forces a graph G, |G| = λ, to havecountable chromatic number, then G has countable chromatic
number.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

8 / 23

Sketch of the proof - 2
Suppose, towards a contradiction, that J 00 G has chromaticbnumber ℵ0 in M[Gb ∗ H].
b we have that G has countable
Since J 00 G ≈ G in V [Gb ∗ H],bchromatic number in V [Gb ∗ H].
b is < λ close, so
The forcing poset to get from V [Gb] to V [Gb ∗ H]
we can apply the following lemma:

Lemma (< λ-closed transfer)
Let D a < λ-closed poset. If D forces a graph G, |G| = λ, to havecountable chromatic number, then G has countable chromatic
number.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

8 / 23

Sketch of the proof - 2
Suppose, towards a contradiction, that J 00 G has chromaticbnumber ℵ0 in M[Gb ∗ H].
b we have that G has countable
Since J 00 G ≈ G in V [Gb ∗ H],bchromatic number in V [Gb ∗ H].
b is < λ close, so
The forcing poset to get from V [Gb] to V [Gb ∗ H]
we can apply the following lemma:

Lemma (< λ-closed transfer)
Let D a < λ-closed poset. If D forces a graph G, |G| = λ, to havecountable chromatic number, then G has countable chromatic
number.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

8 / 23

Sketch of the proof - 2
Suppose, towards a contradiction, that J 00 G has chromaticbnumber ℵ0 in M[Gb ∗ H].
b we have that G has countable
Since J 00 G ≈ G in V [Gb ∗ H],bchromatic number in V [Gb ∗ H].
b is < λ close, so
The forcing poset to get from V [Gb] to V [Gb ∗ H]
we can apply the following lemma:

Lemma (< λ-closed transfer)
Let D a < λ-closed poset. If D forces a graph G, |G| = λ, to havecountable chromatic number, then G has countable chromatic
number.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

8 / 23

Sketch of the proof - 3
We also have that V [Gb] is a κ-centred generic extension of V [G ∗ H],
so we can apply the next lemma:

Lemma (κ-centred transfer)
Let C a κ-centered poset. Suppose that C forces a graph G, |G| > κ,to have countable chromatic number. Then the chromatic number of
G is ≤ κ.
We just proved that in V [G ∗ H], we have Chr(G) ≤ κ. But thiscontradicts the assumption
V [G ∗ H] |= “ Chr(G) = ℵ2 = λ > κ ”

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

9 / 23

Sketch of the proof - 3
We also have that V [Gb] is a κ-centred generic extension of V [G ∗ H],
so we can apply the next lemma:

Lemma (κ-centred transfer)
Let C a κ-centered poset. Suppose that C forces a graph G, |G| > κ,to have countable chromatic number. Then the chromatic number of
G is ≤ κ.
We just proved that in V [G ∗ H], we have Chr(G) ≤ κ. But thiscontradicts the assumption
V [G ∗ H] |= “ Chr(G) = ℵ2 = λ > κ ”

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

9 / 23

Sketch of the proof - 3
We also have that V [Gb] is a κ-centred generic extension of V [G ∗ H],
so we can apply the next lemma:

Lemma (κ-centred transfer)
Let C a κ-centered poset. Suppose that C forces a graph G, |G| > κ,to have countable chromatic number. Then the chromatic number of
G is ≤ κ.
We just proved that in V [G ∗ H], we have Chr(G) ≤ κ. But thiscontradicts the assumption
V [G ∗ H] |= “ Chr(G) = ℵ2 = λ > κ ”

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

9 / 23

Applications in topology

On the same Foreman-Laver model, Tall applied the same
technique to obtain some results about some topologicalproperties.
Instead of transferring down properties from ℵ2 to ℵ1 , his results
go the opposite way, transferring properties from ℵ1 to ℵ2 , orhigher.
For his results, we need some definitions:O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

10 / 23

Applications in topology

On the same Foreman-Laver model, Tall applied the same
technique to obtain some results about some topologicalproperties.
Instead of transferring down properties from ℵ2 to ℵ1 , his results
go the opposite way, transferring properties from ℵ1 to ℵ2 , orhigher.
For his results, we need some definitions:O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

10 / 23

Applications in topology

On the same Foreman-Laver model, Tall applied the same
technique to obtain some results about some topologicalproperties.
Instead of transferring down properties from ℵ2 to ℵ1 , his results
go the opposite way, transferring properties from ℵ1 to ℵ2 , orhigher.
For his results, we need some definitions:O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

10 / 23

Definition
A collection Y of subsets of a topological space is discrete if
each point in the space has a neighbourhood which meets at
most one member of the collection.
A discrete collection Y is separated if there exist pairwise
disjoint open sets UY for each Y ∈ Y such that Y ⊆ UY .
A space is κ-collectionwise Hausdorff if every discrete
collection of size ≤ κ consisting of points (singletons) is
separated.
A space is κ-collectionwise normal if every discrete collectionof ≤ κ many closed sets is separated.
A space is weakly-κ-collectionwise Hausdorff if every
discrete collection of κ points (singletons) has some separated
subcollection of size κ.
O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

11 / 23

Definition
A collection Y of subsets of a topological space is discrete if
each point in the space has a neighbourhood which meets at
most one member of the collection.
A discrete collection Y is separated if there exist pairwise
disjoint open sets UY for each Y ∈ Y such that Y ⊆ UY .
A space is κ-collectionwise Hausdorff if every discrete
collection of size ≤ κ consisting of points (singletons) is
separated.
A space is κ-collectionwise normal if every discrete collectionof ≤ κ many closed sets is separated.
A space is weakly-κ-collectionwise Hausdorff if every
discrete collection of κ points (singletons) has some separated
subcollection of size κ.
O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

11 / 23

Definition
A collection Y of subsets of a topological space is discrete if
each point in the space has a neighbourhood which meets at
most one member of the collection.
A discrete collection Y is separated if there exist pairwise
disjoint open sets UY for each Y ∈ Y such that Y ⊆ UY .
A space is κ-collectionwise Hausdorff if every discrete
collection of size ≤ κ consisting of points (singletons) is
separated.
A space is κ-collectionwise normal if every discrete collectionof ≤ κ many closed sets is separated.
A space is weakly-κ-collectionwise Hausdorff if every
discrete collection of κ points (singletons) has some separated
subcollection of size κ.
O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

11 / 23

Definition
A collection Y of subsets of a topological space is discrete if
each point in the space has a neighbourhood which meets at
most one member of the collection.
A discrete collection Y is separated if there exist pairwise
disjoint open sets UY for each Y ∈ Y such that Y ⊆ UY .
A space is κ-collectionwise Hausdorff if every discrete
collection of size ≤ κ consisting of points (singletons) is
separated.
A space is κ-collectionwise normal if every discrete collectionof ≤ κ many closed sets is separated.
A space is weakly-κ-collectionwise Hausdorff if every
discrete collection of κ points (singletons) has some separated
subcollection of size κ.
O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

11 / 23

Definition
A collection Y of subsets of a topological space is discrete if
each point in the space has a neighbourhood which meets at
most one member of the collection.
A discrete collection Y is separated if there exist pairwise
disjoint open sets UY for each Y ∈ Y such that Y ⊆ UY .
A space is κ-collectionwise Hausdorff if every discrete
collection of size ≤ κ consisting of points (singletons) is
separated.
A space is κ-collectionwise normal if every discrete collectionof ≤ κ many closed sets is separated.
A space is weakly-κ-collectionwise Hausdorff if every
discrete collection of κ points (singletons) has some separated
subcollection of size κ.
O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

11 / 23

Definition (κ-paracompactness)
Recall that a refinement of a cover U of a topological space X isa cover V of X such that for every V ∈ V there is some UV ∈ Usuch that V ⊂ UV .
We say that V is locally finite if every point in the space has aneighbourhood which meets finitely many elements of V.
A topological space is κ-paracompact if every open cover ofsize ≤ κ has a locally finite open refinement.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

12 / 23

Definition (κ-paracompactness)
Recall that a refinement of a cover U of a topological space X isa cover V of X such that for every V ∈ V there is some UV ∈ Usuch that V ⊂ UV .
We say that V is locally finite if every point in the space has aneighbourhood which meets finitely many elements of V.
A topological space is κ-paracompact if every open cover ofsize ≤ κ has a locally finite open refinement.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

12 / 23

Definition (κ-paracompactness)
Recall that a refinement of a cover U of a topological space X isa cover V of X such that for every V ∈ V there is some UV ∈ Usuch that V ⊂ UV .
We say that V is locally finite if every point in the space has aneighbourhood which meets finitely many elements of V.
A topological space is κ-paracompact if every open cover ofsize ≤ κ has a locally finite open refinement.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

12 / 23

Similarly to the proof of the transfer of the chromatic number, Tallproved the following results:

Theorem (Tall)
In the model V [G ∗ H], let X be a first countable topological space.
(a) If X is weakly-ℵ1 -collectionwise Hausdorff, then X is
weakly-ℵ2 -collectionwise Hausdorff.
(b) If X is locally ℵ2 -c.c. and ℵ1 -collectionwise Hausdorff, then X is
ℵ2 -collectionwise Hausdorff.
(c) If X has size ≤ ℵ2 and is normal and hereditarilyℵ1 -paracompact, then X is paracompact.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

13 / 23

Similarly to the proof of the transfer of the chromatic number, Tallproved the following results:

Theorem (Tall)
In the model V [G ∗ H], let X be a first countable topological space.
(a) If X is weakly-ℵ1 -collectionwise Hausdorff, then X is
weakly-ℵ2 -collectionwise Hausdorff.
(b) If X is locally ℵ2 -c.c. and ℵ1 -collectionwise Hausdorff, then X is
ℵ2 -collectionwise Hausdorff.
(c) If X has size ≤ ℵ2 and is normal and hereditarilyℵ1 -paracompact, then X is paracompact.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

13 / 23

Similarly to the proof of the transfer of the chromatic number, Tallproved the following results:

Theorem (Tall)
In the model V [G ∗ H], let X be a first countable topological space.
(a) If X is weakly-ℵ1 -collectionwise Hausdorff, then X is
weakly-ℵ2 -collectionwise Hausdorff.
(b) If X is locally ℵ2 -c.c. and ℵ1 -collectionwise Hausdorff, then X is
ℵ2 -collectionwise Hausdorff.
(c) If X has size ≤ ℵ2 and is normal and hereditarilyℵ1 -paracompact, then X is paracompact.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

13 / 23

Similarly to the proof of the transfer of the chromatic number, Tallproved the following results:

Theorem (Tall)
In the model V [G ∗ H], let X be a first countable topological space.
(a) If X is weakly-ℵ1 -collectionwise Hausdorff, then X is
weakly-ℵ2 -collectionwise Hausdorff.
(b) If X is locally ℵ2 -c.c. and ℵ1 -collectionwise Hausdorff, then X is
ℵ2 -collectionwise Hausdorff.
(c) If X has size ≤ ℵ2 and is normal and hereditarilyℵ1 -paracompact, then X is paracompact.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

13 / 23

Some counterexamples
Example (Rudin-Tall)
Suppose there exists S ⊆ Eωω02 = {α ∈ ω2 : cf(α) = ω0 } which isstationary and does not reflect. Then (c) fails.

Example (Fleissner)Suppose the existence of S like above. Then (b) fails.

Example (LaBergue-Landver)
Suppose ω1 . Then there exists a first countable ℵ1 -collectionwise
Hausdorff space which is not weakly ℵ2 -collectionwise Hausdorff(hence (a) fails).

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

14 / 23

Some counterexamples
Example (Rudin-Tall)
Suppose there exists S ⊆ Eωω02 = {α ∈ ω2 : cf(α) = ω0 } which isstationary and does not reflect. Then (c) fails.

Example (Fleissner)Suppose the existence of S like above. Then (b) fails.

Example (LaBergue-Landver)
Suppose ω1 . Then there exists a first countable ℵ1 -collectionwise
Hausdorff space which is not weakly ℵ2 -collectionwise Hausdorff(hence (a) fails).

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

14 / 23

Some counterexamples
Example (Rudin-Tall)
Suppose there exists S ⊆ Eωω02 = {α ∈ ω2 : cf(α) = ω0 } which isstationary and does not reflect. Then (c) fails.

Example (Fleissner)Suppose the existence of S like above. Then (b) fails.

Example (LaBergue-Landver)
Suppose ω1 . Then there exists a first countable ℵ1 -collectionwise
Hausdorff space which is not weakly ℵ2 -collectionwise Hausdorff(hence (a) fails).

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

14 / 23

Motivation

Going back to the transfer property of the chromatic number, it iseasy to see that:
TrChr (ℵ3 , ℵ2 ) ∧ TrChr (ℵ2 , ℵ1 ) =⇒ TrChr (ℵ3 , ℵ1 )But does the reciprocal holds?
NO, if ZFC +“ there exists a huge cardinal ” is consistent.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

15 / 23

Motivation

Going back to the transfer property of the chromatic number, it iseasy to see that:
TrChr (ℵ3 , ℵ2 ) ∧ TrChr (ℵ2 , ℵ1 ) =⇒ TrChr (ℵ3 , ℵ1 )But does the reciprocal holds?
NO, if ZFC +“ there exists a huge cardinal ” is consistent.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

15 / 23

Motivation

Going back to the transfer property of the chromatic number, it iseasy to see that:
TrChr (ℵ3 , ℵ2 ) ∧ TrChr (ℵ2 , ℵ1 ) =⇒ TrChr (ℵ3 , ℵ1 )But does the reciprocal holds?
NO, if ZFC +“ there exists a huge cardinal ” is consistent.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

15 / 23

New result

TheoremAssuming the existence of a huge cardinal, then there exists W aforcing extension of V such that
W |= “ TrChr (ℵ3 , ℵ1 ) ∧ ¬TrChr (ℵ2 , ℵ1 ) ”

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

16 / 23

Sketch of the construction
Suppose the existence of a huge cardinal κ with target λ.
We can modify the poset P ∗ Ṙ to create P0 ∗ Ṙ0 such thatP0 ∗ Ṙ0 k– “ k = ℵ1 ∧ κ+ = ℵ2 ∧ λ = ℵ3 ∧ TrChr (ℵ3 , ℵ1 ) ”

Let B be the poset created by Baumgartner. By taking
advantage of the small size and countable closure of B, we canconstruct P00 such that P00 ∗ Ṙ0 ∗ Ḃ still forces the above.
So we conclude that
P00 ∗ Ṙ0 ∗ Ḃ k– “ TrChr (ℵ3 , ℵ1 ) ∧ ¬TrChr (ℵ2 , ℵ1 ) ”

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

17 / 23

Sketch of the construction
Suppose the existence of a huge cardinal κ with target λ.
We can modify the poset P ∗ Ṙ to create P0 ∗ Ṙ0 such thatP0 ∗ Ṙ0 k– “ k = ℵ1 ∧ κ+ = ℵ2 ∧ λ = ℵ3 ∧ TrChr (ℵ3 , ℵ1 ) ”

Let B be the poset created by Baumgartner. By taking
advantage of the small size and countable closure of B, we canconstruct P00 such that P00 ∗ Ṙ0 ∗ Ḃ still forces the above.
So we conclude that
P00 ∗ Ṙ0 ∗ Ḃ k– “ TrChr (ℵ3 , ℵ1 ) ∧ ¬TrChr (ℵ2 , ℵ1 ) ”

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

17 / 23

Sketch of the construction
Suppose the existence of a huge cardinal κ with target λ.
We can modify the poset P ∗ Ṙ to create P0 ∗ Ṙ0 such thatP0 ∗ Ṙ0 k– “ k = ℵ1 ∧ κ+ = ℵ2 ∧ λ = ℵ3 ∧ TrChr (ℵ3 , ℵ1 ) ”

Let B be the poset created by Baumgartner. By taking
advantage of the small size and countable closure of B, we canconstruct P00 such that P00 ∗ Ṙ0 ∗ Ḃ still forces the above.
So we conclude that
P00 ∗ Ṙ0 ∗ Ḃ k– “ TrChr (ℵ3 , ℵ1 ) ∧ ¬TrChr (ℵ2 , ℵ1 ) ”

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

17 / 23

Sketch of the construction
Suppose the existence of a huge cardinal κ with target λ.
We can modify the poset P ∗ Ṙ to create P0 ∗ Ṙ0 such thatP0 ∗ Ṙ0 k– “ k = ℵ1 ∧ κ+ = ℵ2 ∧ λ = ℵ3 ∧ TrChr (ℵ3 , ℵ1 ) ”

Let B be the poset created by Baumgartner. By taking
advantage of the small size and countable closure of B, we canconstruct P00 such that P00 ∗ Ṙ0 ∗ Ḃ still forces the above.
So we conclude that
P00 ∗ Ṙ0 ∗ Ḃ k– “ TrChr (ℵ3 , ℵ1 ) ∧ ¬TrChr (ℵ2 , ℵ1 ) ”

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

17 / 23

We can use the same argument with Tall’s result (a) . Instead of B,
we use a poset which forces ω1 . We obtain:

TheoremSuppose the existence of a huge cardinal. Then there exists a forcing
extension W of V in which every first countable weaklyℵ1 -collectionwise Hausdorff space is ℵ3 -collectionwise Hausdorff, butthere exists some first countable ℵ1 -collectionwise Hausdorff which is
not weakly ℵ2 -collectionwise Hausdorff in W .

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

18 / 23

Generalizing Baumgartner’s poset
By changing the parameters on the construction of the poset B by
Baumgartner, we obtain:

TheoremLet κ be regular. Assume that 2κ = κ+ . Then there is a ≤ κ-closed
κ++ -c.c. poset B(κ) of size ≤ κ++ which adds a graph of size andchromatic number κ++ such that all subgraphs of size ≤ κ+ havechromatic number ≤ κ.
In particular, this means that, for any regular κ,the consistency of the
failure of TrChr (κ++ , κ+ ) is consistent.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

19 / 23

Generalizing Baumgartner’s poset
By changing the parameters on the construction of the poset B by
Baumgartner, we obtain:

TheoremLet κ be regular. Assume that 2κ = κ+ . Then there is a ≤ κ-closed
κ++ -c.c. poset B(κ) of size ≤ κ++ which adds a graph of size andchromatic number κ++ such that all subgraphs of size ≤ κ+ havechromatic number ≤ κ.
In particular, this means that, for any regular κ,the consistency of the
failure of TrChr (κ++ , κ+ ) is consistent.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

19 / 23

Generalizing Baumgartner’s poset
By changing the parameters on the construction of the poset B by
Baumgartner, we obtain:

TheoremLet κ be regular. Assume that 2κ = κ+ . Then there is a ≤ κ-closed
κ++ -c.c. poset B(κ) of size ≤ κ++ which adds a graph of size andchromatic number κ++ such that all subgraphs of size ≤ κ+ havechromatic number ≤ κ.
In particular, this means that, for any regular κ,the consistency of the
failure of TrChr (κ++ , κ+ ) is consistent.

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

19 / 23

Generalization of the new result

Using the above generalization together with the previous argument,
we obtain the generalized version:

TheoremSuppose κ is a huge cardinal with target λ. Let α be an ordinal suchthat (ℵα )V < κ is regular. Let also n + 2 < m < ω. Then, there
exists W a generic extension of V such that (ℵα+1 )W = κ,(ℵα+m )W = λ and
W |= “ TrChr (ℵα+m , ℵα+1 ) ∧ ¬TrChr (ℵα+n+2 , ℵα+n+1 ) ”

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

20 / 23

Question
Is it consistent (modulo some large cardinal assumption) that
TrChr (ℵ3 , ℵ1 ) 6=⇒ TrChr (ℵ3 , ℵ2 )?

Question
Can we obtain the consistency of some of these transfer properties
from some weaker large cardinal assumption?

Question
Are these transfer properties consistent with the negation of GCH?

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

21 / 23

Question
Is it consistent (modulo some large cardinal assumption) that
TrChr (ℵ3 , ℵ1 ) 6=⇒ TrChr (ℵ3 , ℵ2 )?

Question
Can we obtain the consistency of some of these transfer properties
from some weaker large cardinal assumption?

Question
Are these transfer properties consistent with the negation of GCH?

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

21 / 23

Question
Is it consistent (modulo some large cardinal assumption) that
TrChr (ℵ3 , ℵ1 ) 6=⇒ TrChr (ℵ3 , ℵ2 )?

Question
Can we obtain the consistency of some of these transfer properties
from some weaker large cardinal assumption?

Question
Are these transfer properties consistent with the negation of GCH?

O. M. Rodriges, André (Kobe University)On some downwards transfer properties on Foreman-Laver
11/29/2016
model

21 / 23

Thank you very much!

References
James E. Baumgartner.
Generic graph construction.
J. Symbolic Logic, 49(1):234–240, 1984.
Matthew Foreman and Richard Laver.
Some downwards transfer properties for ℵ2 .
Adv. in Math., 67(2):230–238, 1988.
Kenneth Kunen.
Saturated ideals.
J. Symbolic Logic, 43(1):65–76, 1978.
Franklin D. Tall.
Topological applications of generic huge embeddings.
Trans. Amer. Math. Soc., 341(1):45–68, 1994.