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?



$\Sigma_1^0-COH$?
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](http://dx.doi.org/10.1002/malq.19930390153), [MR1270396](http://www.ams.org/mathscinet-getitem?mr=1270396)] (that survive the later corrections [Math. Logic Quart. 43 (1997), 569; doi:[10.1002/malq.19970430412](http://dx.doi.org/10.1002/malq.19970430412), [MR1477624](http://www.ams.org/mathscinet-getitem?mr=1477624)]) 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](http://dx.doi.org/10.1007/s00153-012-0297-4), arXiv:[1110.6559](http://arxiv.org/abs/1110.6559), [MR2975428](http://www.ams.org/mathscinet-getitem?mr=2975428)] 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](http://dx.doi.org/10.2178/jsl/1174668391), [MR2298478](http://www.ams.org/mathscinet-getitem?mr=2298478)].


