A set $X$ is called *cohesive for $(R\_i)\_{i\in \mathbb{N}}$* if it is infinite and for each $i$ we have $X\subseteq^\* R\_i$ or $X\subseteq^\* \overline{R\_i}$. (Where $X\subseteq^\*Y$ means that $X$ is contained in $Y$ up to finitely many exceptions.)


A set $X$ is called *cohesive* if it is cohesive for all computable enumerable sets.


Further, a set $X$ is called *1-generic* if for each computably enumerable set $S\subseteq 2^{<\mathbb{N}}$ of strings there is an initial segment of $\sigma$ of $X$ such that either $\sigma \in S$ or $\sigma \not\subseteq \tau$ for all $\tau \in \sigma$.


Both notions are well studies in computability theory. However, I was not able to find a answer to the following question in the literature:


Is (Turing) below each cohesive set a 1-generic set?




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To give some background on the question:
The reverse of the above question is known to be false. That is there is a 1-generic which has no cohesive set below it.
For the proof one has to note that there are low 1-generic sets but no low cohesive sets. (See [Jockush, Stephan: A cohesive set which is not high](http://onlinelibrary.wiley.com/doi/10.1002/malq.19930390153/abstract)).


Moreover it is known that below each cohesive set is a **weakly** 1-generic. (A set $X$ is weakly 1-generic if for each dense c.e. set $S$ there is an inital segment $\sigma$ of $X$ with $\sigma\in S$.)



Is below every cohesive set a 1-generic?
No -- each high degree contains a cohesive set. And Cooper showed there is a high minimal degree. But no 1-generic has minimal degree.


