Finding 1-generic paths through a tree $T \subseteq 2^{<\omega}$

Question

Consider Cantor space $2^\omega$ with the standard topology generated by open sets $[\sigma] = \{ \sigma^\frown x: x \in 2^\omega \}$. If $A \subseteq 2^{<\omega}$ and $x \in 2^\omega$, we say $A$ is dense along $x$ if for every prefix $\sigma \prec x$, there is $\tau \succ \sigma$ such that all finite extensions of $\tau$ are in $A$.

An element $x \in 2^\omega$ is 1-generic if, for every $\Sigma^0_1$ (computably enumerable) set $A \subseteq 2^{<\omega}$ which is dense along $x$, we have $x \in [A]$ ($x$ is a path through $A$). I think this is the standard definition (from here).

Now, suppose $T \subseteq 2^{<\omega}$ is a tree. What conditions can we impose on $T$ that guarantee $[T]$ contains a 1-generic member? Effectively, I'm looking for some type of "generic basis theorem". In particular, if $T$ is infinite and $\Sigma^0_2$, can we guarantee it contains a 1-generic path?

Answer 1

What conditions can we impose on $T$ that guarantee $[T]$ contains a 1-generic member?

An element that is 1-generic relative to $T$ will not be on $[T]$ unless $[T]$ contains a whole clopen cone $[\sigma]$. Since "most" 1-generics are 1-generic relative to $T$, I suppose this means the condition to impose is basically that $[\sigma]\subseteq [T]$ for some $\sigma\in 2^{<\omega}$.

Effectively, I'm looking for some type of "generic basis theorem". In particular, if $T$ is infinite and $\Sigma^0_2$, can we guarantee it contains a 1-generic path?

No, if we let $T$ consist of all diagonally non-recursive $\{0,1\}$-valued functions then $T$ contains no 1-generic path. This is because one can show that no 1-generic computes a DNR function.

Answer 2

I don't think you were asking for 1-genericity relative to $T$ but just plain old normal 1-genericity. I’m going to assume $T$ has no terminal nodes since if it doesn't things get more messy (though I did deal with that way back in my thesis).

The difficulty with any useful basis result here is that you lose if $T$ is too definable. Obviously if T contains a full cone $[\sigma]$ it contains a generic so let's suppose that $\sim T$ is dense (every string can be extended to meet it). But now if $T$ is $\Pi^0_1$ (and hence also if it is computable) it fails to have any generic paths since T complement itself is the witnessing $\Sigma^0_1$ set. But a really complex T need nor help either.

The best I think you can do for a general answer is the obvious thing you would start with: if $\sigma \in T$ and W is a $\Sigma^0_1$ set then you need an extension of $\sigma$ in $T$ that either meets With or strongly avoids that extension. But that's just another way of stating the genericity requirement. You can probably hide that a bit better but I don't think there are any useful basis type results here.