TAOCP 7.2.2.2 Exercise 1
The shortest satisfiable set of clauses is the empty set of clauses, $F=\varnothing$.
Section 7.2.2.2: Satisfiability
Exercise 1. [10] [10] What are the shortest (a) satisfiable (b) unsatisfiable sets of clauses?
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The shortest satisfiable set of clauses is the empty set of clauses, $F=\varnothing$. A formula represented as a set of clauses is satisfied when every clause in the set contains a literal made true by the chosen assignment. Since $\varnothing$ contains no clauses, every assignment satisfies it. Therefore no shorter clause set exists, because a set cannot have fewer than zero elements.
The shortest unsatisfiable set of clauses is $F={\epsilon}$, where $\epsilon$ is the empty clause. The empty clause contains no literals, so there is no assignment that can satisfy it. A set with no clauses is satisfiable, as shown above, so an unsatisfiable set must contain at least one clause. Hence ${\epsilon}$ has the minimum possible size and is the shortest unsatisfiable clause set.
Thus the answers are
$\boxed{\text{(a) }\varnothing \qquad \text{(b) }{\epsilon}}$