To obtain a minimal unsatisfiable 3-SAT example, we start with a two-clause contradiction representing \( u_1 \wedge \neg u_1 \) as the base case. This base case is not 3-SAT. We then grow the clause as follows: suppose \( C_{i-1} \) containing \( i-1 \) variables is unsatisfiable, then let \( C_i = \left\{ c \vee u_i, c \vee \neg u_i \mid c \in C_{i-1} \right\} \), and \( C_i \) is also unsatisfiable.

To carry out the algorithm, we start with the base case.

- \( u_1 \)
- \( \neg u_1 \)

Then we add another variable to each clause:

- For \( u_1 \):
- \( u_1 \vee u_2 \)
- \( u_1 \vee \neg u_2 \)
- For \( \neg u_1 \):
- \( \neg u_1 \vee u_2 \)
- \( \neg u_1 \vee \neg u_2 \)

Then we add another variable to each clause:

- For \( u_1 \vee u_2 \)
- \( u_1 \vee u_2 \vee u_3 \)
- \( u_1 \vee u_2 \vee \neg u_3 \)
- For \( u_1 \vee \neg u_2 \)
- \( u_1 \vee \neg u_2 \vee u_3 \)
- \( u_1 \vee \neg u_2 \vee \neg u_3 \)
- For \( \neg u_1 \vee u_2 \)
- \( \neg u_1 \vee u_2 \vee u_3 \)
- \( \neg u_1 \vee u_2 \vee \neg u_3 \)
- For \( \neg u_1 \vee \neg u_2 \)
- \( \neg u_1 \vee \neg u_2 \vee u_3 \)
- \( \neg u_1 \vee \neg u_2 \vee \neg u_3 \)

And here we have an unsatisfiable 3SAT example with 3 variables and 8 clauses.