The number of clauses in an unsatisfiable CNF
For arbitrarily large number $n$ of variables, the following CNF formula $\phi$ is not satisfiable, has only three clauses, and a $2K_2$-free clause-variable incidence graph:
$C_1=(x_1)$, $C_2=(\neg x_1)$, $C_3=(x_1,\ldots,x_n)$.
Thus, to get more interesting lower bounds one needs to make assumption about the minimum clause size or about the size of classes of clauses that contain the same variables (i.e. that have identical neighborhoods in the clause-variable incidence graph).
Observe that when the minimum clause size is $k$, then the example above can be adapted to give formulas with $2^k+1$ clauses and arbitrarily large number of variables $n$ that are not satisfiable. Moreover, when the number of clauses is $m&lt;2^k+1$, then the formula is satisfiable even for arbitrarily large $n&gt;k$: Let $C_m$ be the clause that has is incident with the largest set of variables. Since $n&gt;k$, there is a variable $y$ that is not contained in the variable set of $C_1$, the clause with the smallest set of incident variables. Now, as for $k$-CNF formulas, the clauses $C_1,\ldots,C_{m-1}$ can be satisfied via an assignment of the first $k$ variables (which they all contain) and $C_m$ can be satisfied via the appropriate assignment of $y$.