Consider the following properties for a subset $A$ of $\mathbb{N}$:

(1) $A$ is large: $\Sigma_{n \in A}$$ 1\over n$$=\infty,$

(2) $A^\infty=Limsup$$ |A \cap \{ 1, \dots, n\}|\over n >0$,

(3) $A_\infty=Liminf$$ |A \cap \{ 1, \dots, n\}|\over n >0.$

By a conjecture of Erdos-Turan, if $A$ is large, then it contains arithmetic progressions of any given (finite) length.

By a result of Szemerédi, if $A^\infty >0,$ then $A$ contains infinite arithmetic progressions of length $k$ for all positive integers $k$.

Let's consider these properties for generic subsets of $\omega$ added by forcing (I assume they do not contain $0$) and let me say a few examples:

(a) If $r$ is Cohen, then $r$ is large, $r^\infty=1, r_\infty=0$ and for any $K,L$, we can find $M$ such that $M, M+L, M+2L, \dots M+KL$ are in $r$.

(b) If $r$ is Random, then $r$ is large, and $r^\infty=r_\infty=$$1\over 2.$ So by Szemerédi's result, it contains arbitrary large arithmetic progressions.

Question 1. Is there a direct proof, without using of Szemerédi's result, that $r$ contains arbitrary large arithmetic progressions?

(c) If $r$ is Mathias, related to an ultrafilter $U$, then the properties of $r$ depends on $U$.

It is possible to say the same results for some other generic reals, but as there are many generic reals that I do not know, I would like to ask a more general question:

Question 2. Suppose that $r$ is a generic real added by forcing (we assume it does not contain $0$). Disuss if $r$ is large, and what are $r^\infty$ and $r_\infty$. Also say if $r$ contains arbitrary large arithmetic progressions or not (I would rather direct proofs instead of referring to some known results (say for example, as $r^\infty>0,$ it contains arbitrary large arithmetic progressions)).

Consider the following properties for a subset $A$ of $\mathbb{N}$:

(1) $A$ is large: $\sum_{n \in A}$$ 1\over n$$=\infty,$

(2) $A^\infty=\limsup \frac{|A \cap \{ 1, \dots, n\}|}{n} >0$,

(3) $A_\infty=\liminf \frac{|A \cap \{ 1, \dots, n\}|}{n} >0.$

By a conjecture of Erdős-Turán, if $A$ is large, then it contains arithmetic progressions of any given (finite) length.

By a result of Szemerédi, if $A^\infty >0,$ then $A$ contains infinite arithmetic progressions of length $k$ for all positive integers $k$.

Let's consider these properties for generic subsets of $\omega$ added by forcing (I assume they do not contain $0$) and let me say a few examples:

(a) If $r$ is Cohen, then $r$ is large, $r^\infty=1, r_\infty=0$ and for any $K,L$, we can find $M$ such that $M, M+L, M+2L, \dots M+KL$ are in $r$.

(b) If $r$ is Random, then $r$ is large, and $r^\infty=r_\infty=$$1\over 2.$ So by Szemerédi's result, it contains arbitrary large arithmetic progressions.

Question 1. Is there a direct proof, without using of Szemerédi's result, that $r$ contains arbitrary large arithmetic progressions?

(c) If $r$ is Mathias, related to an ultrafilter $U$, then the properties of $r$ depends on $U$.

It is possible to say the same results for some other generic reals, but as there are many generic reals that I do not know, I would like to ask a more general question:

Question 2. Suppose that $r$ is a generic real added by forcing (we assume it does not contain $0$). Disuss if $r$ is large, and what are $r^\infty$ and $r_\infty$. Also say if $r$ contains arbitrary large arithmetic progressions or not (I would rather direct proofs instead of referring to some known results (say for example, as $r^\infty>0,$ it contains arbitrary large arithmetic progressions)).

Consider the following properties for a subset $A$ of $\mathbb{N}$:

(1) $A$ is large: $\Sigma_{n \in A}$$ 1\over n$$=\infty,$

(2) $A^\infty=Limsup$$ |A \cap \{ 1, \dots, n\}|\over n >0$,

(3) $A_\infty=Liminf$$ |A \cap \{ 1, \dots, n\}|\over n >0.$

By a conjecture of Erdos-Turan, if $A$ is large, then it contains arithmetic progressions of any given (finite) length.

By a result of Szemerédi, if $A^\infty >0,$ then $A$ contains infinite arithmetic progressions of length $k$ for all positive integers $k$.

Let's consider these properties for generic subsets of $\omega$ added by forcing (I assume they do not contain $0$) and let me say a few examples:

(a) If $r$ is Cohen, then $r$ is large, $r^\infty=1, r_\infty=0$ and for any $K,L$, we can find $M$ such that $M, M+L, M+2L, \dots M+KL$ are in $r$.

(b) If $r$ is Random, then $r$ is large, and $r^\infty=r_\infty=$$1\over 2.$ So by Szemerédi's result, it contains arbitrary large arithmetic progressions.

Question 1. Is there a direct proof, without using of Szemerédi's result, that $r$ contains arbitrary large arithmetic progressions?

(c) If $r$ is Mathias, related to an ultrafilter $U$, then the properties of $r$ depends on $U$.

It is possible to say the same results for some other generic reals, but as there are many generic reals that I do not know, I would like to ask a more general question:

Question 2. Suppose that $r$ is a generic real added by forcing (we assume it does not contain $0$). Disuss if $r$ is large, and what are $r^\infty$ and $r_\infty$. Also say if $r$ contains arbitrary large arithmetic progressions or not (I would rather direct proofs instead of referring to some known results (say for example, as $r^\infty>0,$ it contains arbitrary large arithmetic progressions)).

The following quote from Shelah shows why I did not check the above for all the reals by myself: "Mathematics is large and time is short and the workers are lazy, so I haven't try it"

Consider the following properties for a subset $A$ of $\mathbb{N}$:

(1) $A$ is large: $\Sigma_{n \in A}$$ 1\over n$$=\infty,$

(2) $A^\infty=Limsup$$ |A \cap \{ 1, \dots, n\}|\over n >0$,

(3) $A_\infty=Liminf$$ |A \cap \{ 1, \dots, n\}|\over n >0.$

By a conjecture of Erdos-Turan, if $A$ is large, then it contains arithmetic progressions of any given (finite) length.

By a result of Szemerédi, if $A^\infty >0,$ then $A$ contains infinite arithmetic progressions of length $k$ for all positive integers $k$.

Let's consider these properties for generic subsets of $\omega$ added by forcing (I assume they do not contain $0$) and let me say a few examples:

(a) If $r$ is Cohen, then $r$ is large, $r^\infty=1, r_\infty=0$ and for any $K,L$, we can find $M$ such that $M, M+L, M+2L, \dots M+KL$ are in $r$.

(b) If $r$ is Random, then $r$ is large, and $r^\infty=r_\infty=$$1\over 2.$ So by Szemerédi's result, it contains arbitrary large arithmetic progressions.

Question 1. Is there a direct proof, without using of Szemerédi's result, that $r$ contains arbitrary large arithmetic progressions?

(c) If $r$ is Mathias, related to an ultrafilter $U$, then the properties of $r$ depends on $U$.

It is possible to say the same results for some other generic reals, but as there are many generic reals that I do not know, I would like to ask a more general question:

Question 2. Suppose that $r$ is a generic real added by forcing (we assume it does not contain $0$). Disuss if $r$ is large, and what are $r^\infty$ and $r_\infty$. Also say if $r$ contains arbitrary large arithmetic progressions or not (I would rather direct proofs instead of referring to some known results (say for example, as $r^\infty>0,$ it contains arbitrary large arithmetic progressions)).