What is the best-known inapproximability result for MIN-3CNF-DELETION? I am really curious what the best-known inapproximability result is for MIN-3CNF-DELETION. To clarify, this is the problem of minimizing the number of unsatisfied clauses in a CNF
SAT formula with at most 3 literals per clause. The objective function is the number of unsatisfied clauses. The strongest hardness result I have been able to find is that of MIN-HORN-DELETION, which is a restricted variant of MIN-3CNF-DELETION. As far as I know the best-known inapproximability result for MIN-HORN-DELETION is that which is given in The Approximability of Constraint Satisfaction Problems by Khanna et al (2000). 

However, I have not yet been able to find explicit references to MIN-3CNF-DELETION. Is it even harder than MIN-HORN-DELETION? In which case, what is the best known inapproximability threshhold? Any feedback would be most welcome. Thanks!
There is no approximation algorithm for MIN-3CNF-DELETION at all since it is not even possible to determine in polynomial time whether a 3 CNF SAT instance is satisfiable or not (if P≠NP). That is, it is not possible to determine if the objective function is equal to 0 or strictly positive. Specifically, if we had an $f(n)$ approximation algorithm for the problem, given a satisfiable instance the algorithm would find a solution of cost $0 \cdot f(n) = 0$ ; that is, a solution that satisfies the instance. 

In contrast, if an instance of Min-Horn-Deletion or Min-2CNF-Deletion is (completely) satisfiable, we can efficiently find a solution that satisfies all clauses. Thus it makes sense to talk about approximation algorithms for Min-Horn-Deletion and Min-2CNF-Deletion.