To solve the problem, we need to analyze the situation where every person seated around a round table calls their left neighbor a liar. We have to identify the condition in terms of the number of people, \(n\), such that the statements hold true.

Let's consider the implications of the statements:

1. If a person calls their left neighbor a liar and is telling the truth, then their left neighbor must indeed be a liar.
2. If a person calls their left neighbor a liar and is lying, then their left neighbor must be a truth-teller, as they cannot both be lying about the same thing.

Given the setup, this establishes a pattern: person A calls person B (to their left) a liar, which implies the following sequence:

• Person A = Truth-teller, Person B = Liar
• Person B = Liar, Person C = Truth-teller

This alternating pattern must continue around the circle. For this arrangement to be consistent in a full circle:

• There must be an even number of people (\(n\) is even), so the cycle can repeat exactly.
• This pattern will result in every alternate person being a liar.

Now, let's evaluate the options:

• All are liars: This is impossible because, in such a case, each person would be lying about their left neighbor, which would create a contradiction.
• \(n\) must be even and every alternate person is a liar: This is consistent with our analysis.
• \(n\) must be odd and every alternate person is a liar: This would not complete the alternate pattern accurately.
• \(n\) must be a prime: Prime numbers do not determine the pattern of liars; rather, it is parity (odd/even) that determines consistent results.

Therefore, the correct answer is: \(n\) must be even and every alternate person is a liar.