$\Sigma_1^0-COH$? In reverse mathematics, $COH$ is a statement that there is a cohesive set for any uniform array of sets. Here uniform array of sets means that there exists a set $B$ such that $x\in B_e \leftrightarrow  (e,x)\in B$, while a set $A$ is cohesive for $\lbrace B_e: e\in \omega\rbrace$ if and only if $A\subset^* B_e$ or $A\subset^*\bar B_e$ for any $e\in \omega$.
I am thinking of a stronger version of $COH:$ replace the uniform family of sets by $\Sigma^0_1$ family, namely there exists a set $B$ such that $x\in B_e \leftrightarrow  \exists z (e,x,z)\in B$ and a cohesive for $\lbrace B_e: e\in \omega\rbrace$ exists. I am searching for literatures that studied this principle previously but so far have not found anything interesting. Does anyone know what has been done regarding this principle?
 A: I also don't recall this principle being directly adressed in the reverse math literature. However, known results can be pieced together to paint a decent picture for $\Sigma^0_1$-COH. I will add to this answer if I find something more.
The results of Jockusch and Stephan, A cohesive set which is not high
[Math. Logic Quart. 39 (1993), 515–530; doi:10.1002/malq.19930390153, MR1270396] (that survive the later corrections [Math. Logic Quart. 43 (1997), 569; doi:10.1002/malq.19970430412, MR1477624]) are very relevant, at least in the context of $\omega$-models. In particular, it follows from these results that $\Sigma^0_1$-COH does not imply arithmetic comprehension. (Actually, it follows from some of my results in A variant of Mathias forcing that preserves $\text{ACA}_0$ 
[Arch. Math. Logic 51 (2012), 751–780; doi:10.1007/s00153-012-0297-4, arXiv:1110.6559, MR2975428] that $\Gamma$-COH does not imply arithmetic comprehension for any class of formulas $\Gamma$.)
The analysis of Jockush and Stephan actually allows us to characterize $\Sigma^0_1$-COH more precisely. Namely, $\Sigma^0_1$-COH is equivalent to the statement that: 


*

*For every set $A$, there is a set $C$ such that either $C \subseteq^* X$ or $C \subseteq^* \bar{X}$ for every $A$-computable set $X$. 


Further thought shows that $\Sigma^0_1$-COH is equivalent to the conjunction of COH with: 


*

*For every set $A$, there are an infinite set $D$ and a sequence $\langle R_k \rangle$ of subsets of $D$ such that if $X$ is $A$-computable then there is a $k$ such that $X \cap D = R_k$


Or with the slight strengthening:


*

*For every set $A$, there are an infinite set $D$ and a sequence $\langle R_k \rangle$ of subsets of $D$ such that for all $i,j$, if $W^A_i \cap W^A_j \cap D = \varnothing$, then there is a $k$ such that $W^A_i \cap D \subseteq R_k$ and $W^A_j \cap D \subseteq \bar{R}_k$.


Note that if we require $D = \omega$, this statement last is equivalent to the Weak König Lemma. Therefore COH and $\Sigma^0_1$-COH are equivalent modulo $\text{WKL}_0$ (or even $\text{RCA}_0+\text{DNR}$).
I don't know whether COH and $\Sigma^0_1$-COH are equivalent over $\text{RCA}_0$, which boils down to asking whether COH proves the bulleted statements above. There is some evidence that suggests that COH might not imply $\Sigma^0_1$-COH over $\text{RCA}_0$. Namely, COH does not imply the uniform version of the last bulleted statement (where $k$ can be effectively computed from $i,j$) by a conservation result of Hirschfeldt and Shore from Combinatorial principles weaker than Ramsey's theorem for pairs [J. Symbolic Logic 72 (2007), 171–206; doi:10.2178/jsl/1174668391, MR2298478].
