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Assuming I understand the definitions correctly, I can give you a couple of references.

(1) Dordal (see below) gives a model in which $\mathfrak{b}=\mathfrak{c}=\aleph_2$ and all towers have cardinality $\aleph_1$. Thus, in his model $\mathfrak{t}=\hat{\mathfrak{t}}=\aleph_1<\aleph_2=\mathfrak{b}=\mathfrak{c}$, and so the last question in your first problem has a negative answer.

Edit: In fact, $\mathfrak{h}=\aleph_2$ in the model since he is using Mathias forcing, so I'm not sure of any good candidate for a non-trivial lower bound for $\hat{\mathfrak{t}}.$

(2) On the other hand, the model of Blass and Shelah discussed here gives a model in which $\mathfrak{t}=\mathfrak{b}=\aleph_1$ and there is a tower of length $\aleph_2$ (namely, the generating tower for the simple $P_{\aleph_2}$-point). Thus, this gives a model where $\mathfrak{t}<\hat{\mathfrak{t}}$.

Edit 3/17/20

I tracked down another reference (due to Dordal [2]) with some more information on this question. Theorem 1.3 in the paper gives a connection between towers in the structures $^\omega\omega$ and $[\omega]^\omega$:

Theorem (Dordal, Theorem 1.3 of [2])

Let $\kappa$ be an uncountable regular cardinal, and suppose there is a $\kappa$-tower in $^\omega\omega$ but not $[\omega]^\omega$. Then there is a $\kappa$-scale in $^\omega\omega$ (that is, $\mathfrak{b}=\mathfrak{d}=\kappa$).

With regard to Problem 1, we may say that if $\mathfrak{b}<\mathfrak{d}$ then $\mathfrak{b}\leq\hat{\mathfrak{t}}$ (apply the above theorem, and use the fact that there is a tower of length $\mathfrak{b}$ in the structure $^\omega\omega$. On the other hand, it is consistent that $\mathfrak{b}=\mathfrak{d}=\aleph_2$ and $\hat{\mathfrak{t}}=\aleph_1$ (this occurs in the model from [1]).

The paper has many other forcing constructions showing that not much more can be said.

[1] Dordal, Peter Lars, A model in which the base-matrix tree cannot have cofinal branches, J. Symb. Log. 52, 651-664 (1987). ZBL0637.03049.

[2] Dordal, Peter Lars, Towers in ([\omega]^{\omega}) and (^{\omega}\omega), Ann. Pure Appl. Logic 45, No. 3, 247-276 (1989). ZBL0686.03024.

Assuming I understand the definitions correctly, I can give you a couple of references.

(1) Dordal (see below) gives a model in which $\mathfrak{b}=\mathfrak{c}=\aleph_2$ and all towers have cardinality $\aleph_1$. Thus, in his model $\mathfrak{t}=\hat{\mathfrak{t}}=\aleph_1<\aleph_2=\mathfrak{b}=\mathfrak{c}$, and so the last question in your first problem has a negative answer.

Edit: In fact, $\mathfrak{h}=\aleph_2$ in the model since he is using Mathias forcing, so I'm not sure of any good candidate for a non-trivial lower bound for $\hat{\mathfrak{t}}.$

(2) On the other hand, the model of Blass and Shelah discussed here gives a model in which $\mathfrak{t}=\mathfrak{b}=\aleph_1$ and there is a tower of length $\aleph_2$ (namely, the generating tower for the simple $P_{\aleph_2}$-point). Thus, this gives a model where $\mathfrak{t}<\hat{\mathfrak{t}}$.

Edit 3/17/20

I tracked down another reference (due to Dordal [2]) with some more information on this question. Theorem 1.3 in the paper gives a connection between towers in the structures $^\omega\omega$ and $[\omega]^\omega$:

Theorem (Dordal, Theorem 1.3 of [2])

Let $\kappa$ be an uncountable regular cardinal, and suppose there is a $\kappa$-tower in $^\omega\omega$ but not $[\omega]^\omega$. Then there is a $\kappa$-scale in $^\omega\omega$ (that is, $\mathfrak{b}=\mathfrak{d}=\kappa$).

With regard to Problem 1, we may say that if $\mathfrak{b}<\mathfrak{d}$ then $\mathfrak{b}\leq\hat{\mathfrak{t}}$ (apply the above theorem, and use the fact that there is a tower of length $\mathfrak{b}$ in the structure $^\omega\omega$. On the other hand, it is consistent that $\mathfrak{b}=\mathfrak{d}=\aleph_2$ and $\hat{\mathfrak{t}}=\aleph_1$ (this occurs in the model from [1]).

The paper has many other forcing constructions showing that not much more can be said.

[1] Dordal, Peter Lars, A model in which the base-matrix tree cannot have cofinal branches, J. Symb. Log. 52, 651-664 (1987). ZBL0637.03049.

[2] Dordal, Peter Lars, Towers in $[\omega]^{\omega}$ and $^{\omega}\omega$, Ann. Pure Appl. Logic 45, No. 3, 247-276 (1989). ZBL0686.03024.

Assuming I understand the definitions correctly, I can give you a couple of references.

(1) Dordal (see below) gives a model in which $\mathfrak{b}=\mathfrak{c}=\aleph_2$ and all towers have cardinality $\aleph_1$. Thus, in his model $\mathfrak{t}=\hat{\mathfrak{t}}=\aleph_1<\aleph_2=\mathfrak{b}=\mathfrak{c}$, and so the last question in your first problem has a negative answer.

Edit: In fact, $\mathfrak{h}=\aleph_2$ in the model since he is using Mathias forcing, so I'm not sure of any good candidate for a non-trivial lower bound for $\hat{\mathfrak{t}}.$

(2) On the other hand, the model of Blass and Shelah discussed here gives a model in which $\mathfrak{t}=\mathfrak{b}=\aleph_1$ and there is a tower of length $\aleph_2$ (namely, the generating tower for the simple $P_{\aleph_2}$-point). Thus, this gives a model where $\mathfrak{t}<\hat{\mathfrak{t}}$.

Edit:

I tracked down another reference (due to Dordal [2]) with some more information on this question. Theorem 1.3 in the paper gives a connection between towers in the structures $^\omega\omega$ and $[\omega]^\omega$:

Theorem (Dordal, Theorem 1.3 of [2])

Let $\kappa$ be an uncountable regular cardinal, and suppose there is a $\kappa$-tower in $^\omega\omega$ but not $[\omega]^\omega$. Then there is a $\kappa$-scale in $^\omega\omega$ (that is, $\mathfrak{b}=\mathfrak{d}=\kappa$).

With regard to Problem 1, we may say that if $\mathfrak{b}<\mathfrak{d}$ then $\mathfrak{b}\leq\hat{\mathfrak{t}}$ (apply the above theorem, and use the fact that there is a tower of length $\mathfrak{b}$ in the structure $^\omega\omega$. On the other hand, it is consistent that $\mathfrak{b}=\mathfrak{d}=\aleph_2$ and $\hat{\mathfrak{t}}=\aleph_1$ (this occurs in the model from [1]).

The paper has many other forcing constructions showing that not much more can be said.

[1] Dordal, Peter Lars, A model in which the base-matrix tree cannot have cofinal branches, J. Symb. Log. 52, 651-664 (1987). ZBL0637.03049.

[2] Dordal, Peter Lars, Towers in ([\omega]^{\omega}) and (^{\omega}\omega), Ann. Pure Appl. Logic 45, No. 3, 247-276 (1989). ZBL0686.03024.

Assuming I understand the definitions correctly, I can give you a couple of references.

(1) Dordal (see below) gives a model in which $\mathfrak{b}=\mathfrak{c}=\aleph_2$ and all towers have cardinality $\aleph_1$. Thus, in his model $\mathfrak{t}=\hat{\mathfrak{t}}=\aleph_1<\aleph_2=\mathfrak{b}=\mathfrak{c}$, and so the last question in your first problem has a negative answer.

Edit: In fact, $\mathfrak{h}=\aleph_2$ in the model since he is using Mathias forcing, so I'm not sure of any good candidate for a non-trivial lower bound for $\hat{\mathfrak{t}}.$

(2) On the other hand, the model of Blass and Shelah discussed here gives a model in which $\mathfrak{t}=\mathfrak{b}=\aleph_1$ and there is a tower of length $\aleph_2$ (namely, the generating tower for the simple $P_{\aleph_2}$-point). Thus, this gives a model where $\mathfrak{t}<\hat{\mathfrak{t}}$.

Edit 3/17/20

I tracked down another reference (due to Dordal [2]) with some more information on this question. Theorem 1.3 in the paper gives a connection between towers in the structures $^\omega\omega$ and $[\omega]^\omega$:

Theorem (Dordal, Theorem 1.3 of [2])

Let $\kappa$ be an uncountable regular cardinal, and suppose there is a $\kappa$-tower in $^\omega\omega$ but not $[\omega]^\omega$. Then there is a $\kappa$-scale in $^\omega\omega$ (that is, $\mathfrak{b}=\mathfrak{d}=\kappa$).

With regard to Problem 1, we may say that if $\mathfrak{b}<\mathfrak{d}$ then $\mathfrak{b}\leq\hat{\mathfrak{t}}$ (apply the above theorem, and use the fact that there is a tower of length $\mathfrak{b}$ in the structure $^\omega\omega$. On the other hand, it is consistent that $\mathfrak{b}=\mathfrak{d}=\aleph_2$ and $\hat{\mathfrak{t}}=\aleph_1$ (this occurs in the model from [1]).

The paper has many other forcing constructions showing that not much more can be said.

[1] Dordal, Peter Lars, A model in which the base-matrix tree cannot have cofinal branches, J. Symb. Log. 52, 651-664 (1987). ZBL0637.03049.

[2] Dordal, Peter Lars, Towers in ([\omega]^{\omega}) and (^{\omega}\omega), Ann. Pure Appl. Logic 45, No. 3, 247-276 (1989). ZBL0686.03024.

Assuming I understand the definitions correctly, I can give you a couple of references.

(1) Dordal (see below) gives a model in which $\mathfrak{b}=\mathfrak{c}=\aleph_2$ and all towers have cardinality $\aleph_1$. Thus, in his model $\mathfrak{t}=\hat{\mathfrak{t}}=\aleph_1<\aleph_2=\mathfrak{b}=\mathfrak{c}$, and so the last question in your first problem has a negative answer.

Edit: In fact, $\mathfrak{h}=\aleph_2$ in the model since he is using Mathias forcing, so I'm not sure of any good candidate for a non-trivial lower bound for $\hat{\mathfrak{t}}.$

(2) On the other hand, the model of Blass and Shelah discussed here gives a model in which $\mathfrak{t}=\mathfrak{b}=\aleph_1$ and there is a tower of length $\aleph_2$ (namely, the generating tower for the simple $P_{\aleph_2}$-point). Thus, this gives a model where $\mathfrak{t}<\hat{\mathfrak{t}}$.

Dordal, Peter Lars, A model in which the base-matrix tree cannot have cofinal branches, J. Symb. Log. 52, 651-664 (1987). ZBL0637.03049.

Assuming I understand the definitions correctly, I can give you a couple of references.

(1) Dordal (see below) gives a model in which $\mathfrak{b}=\mathfrak{c}=\aleph_2$ and all towers have cardinality $\aleph_1$. Thus, in his model $\mathfrak{t}=\hat{\mathfrak{t}}=\aleph_1<\aleph_2=\mathfrak{b}=\mathfrak{c}$, and so the last question in your first problem has a negative answer.

Edit: In fact, $\mathfrak{h}=\aleph_2$ in the model since he is using Mathias forcing, so I'm not sure of any good candidate for a non-trivial lower bound for $\hat{\mathfrak{t}}.$

(2) On the other hand, the model of Blass and Shelah discussed here gives a model in which $\mathfrak{t}=\mathfrak{b}=\aleph_1$ and there is a tower of length $\aleph_2$ (namely, the generating tower for the simple $P_{\aleph_2}$-point). Thus, this gives a model where $\mathfrak{t}<\hat{\mathfrak{t}}$.

Edit:

I tracked down another reference (due to Dordal [2]) with some more information on this question. Theorem 1.3 in the paper gives a connection between towers in the structures $^\omega\omega$ and $[\omega]^\omega$:

Theorem (Dordal, Theorem 1.3 of [2])

Let $\kappa$ be an uncountable regular cardinal, and suppose there is a $\kappa$-tower in $^\omega\omega$ but not $[\omega]^\omega$. Then there is a $\kappa$-scale in $^\omega\omega$ (that is, $\mathfrak{b}=\mathfrak{d}=\kappa$).

With regard to Problem 1, we may say that if $\mathfrak{b}<\mathfrak{d}$ then $\mathfrak{b}\leq\hat{\mathfrak{t}}$ (apply the above theorem, and use the fact that there is a tower of length $\mathfrak{b}$ in the structure $^\omega\omega$. On the other hand, it is consistent that $\mathfrak{b}=\mathfrak{d}=\aleph_2$ and $\hat{\mathfrak{t}}=\aleph_1$ (this occurs in the model from [1]).

The paper has many other forcing constructions showing that not much more can be said.

[1] Dordal, Peter Lars, A model in which the base-matrix tree cannot have cofinal branches, J. Symb. Log. 52, 651-664 (1987). ZBL0637.03049.

[2] Dordal, Peter Lars, Towers in ([\omega]^{\omega}) and (^{\omega}\omega), Ann. Pure Appl. Logic 45, No. 3, 247-276 (1989). ZBL0686.03024.