Liars 1

Let us number the people around the table from $1$ to $14$ . Let $t_1, t_2, \ldots, t_k$ be the positions of the $k$ truth tellers.
Without loss of generality, $t_1=1$ .

Notice that we cannot have three liars in a row, thus $t_{i+1} \leq t_{i}+3$ for all $i$ . Thus $t_k \leq 3k -2.$

But also $t_k \geq 12$ , or there will be at least three consecutive liars after the last truth-teller before we reach the first truth teller again.

So $12 \leq 3k -2$ , i.e. $3k \geq 14$ , or $k \geq 14/3$ .

Since $k$ is an integer, $k \geq 5$ . There must be at least $5$ truth-tellers, so there can be at most $9$ liars.

This bound of $9$ can be reached with $t_i = 3i - 2$ .