I am interested in generalisations of the following observation:
An unsatisfiable $k$-CNF has at least $2^k$ clauses.
A special case of the observation is when $k=n$, where $n$ is the number of variables. In this case, the clause-variable incidence graph is a complete bipartite graph.
I am particularly interested in the question:
Is there a lower bound on the number of clauses in a CNF $\phi$, in terms of the number of variables of $\phi$, when the clause-variable incidence graph of $\phi$ is $2K_2$-free (i.e. is a bipartite chain graph)?
But I am also interested in any generalisation of that question, or references to papers on questions about CNFs with this extremal flavour.
EDIT: Thanks to Christian Komusiewicz for highlighting that my question was ambiguous. The lower bound I'm after should be a function of the number of variables.