This question is regarding MAX-E3SAT problem: Given a set of clauses with exactly three literals, find the maximum number of clauses that can be satisfied. The clauses are expressed as disjunctions of three literals. The literals are variables $x_i\in\{0,1\}$. Assume that the literals are all distinct in a clause, and no clause has both $x_i$ and $\neg x_i$ as the clause will be trivially satisfied. Furthermore, all clauses are considered to be distinct.
Let's say at most a fraction $\alpha$ of the $m$ clauses are satisfiable. Let $k=\sum_i^n (g_i-1)$, where $n$ is the number of variables and $g_i=\text{max}(\{\text{occurrences of }x_i, \text{occurrences of }\neg x_i\})$.
I want to find lower and upper bounds on the ratio $\frac{m}{k}$ given $\alpha$. The bounds can be in terms of $\alpha$.
In my attempt I realized that for an instance of E3SAT, with $l$ variables, to be not satisfiable we need $2^l$ clauses at the minimum. So reversing this, if we have $m$ clauses, then the instance is unsatisfiable if it has $< \log_2(m)$ variables. However, I can't seem to find a way to get a constant bound, and carry forward the intuition to any fraction $\alpha$ of the clauses not being satisfiable.
Edit: I figured a rather conservative and simple bound for $k$. Number of occurrences of all the variables in the instance is $3m$. Then $k< 3m$. However, this seems to be a very conservative bound.
