Problem 10.7. On an island, there are 23 knights and 200 liars; all residents have different names. A tourist who knows this asked each of the 223 residents to write down 200 names of liars on a piece of paper. Each knight correctly wrote down 200 names of liars, while each liar wrote down an arbitrary list of 200 names, which definitely does not include their own name. What is the maximum number of liars the tourist can definitely identify from these data?
Answer: 16.

Solution. First, let's outline a strategy for the liars that allows identifying no more than 16 of them. Let's call the knights $R_{1}, R_{2}, \ldots, R_{23}$, and the liars $-L_{1}, L_{2}, \ldots, L_{200}$. All knights will list $L_{1}, L_{2}, \ldots, L_{200}$ in their lists. For each integer $0 \leqslant i \leqslant 7$, the group of liars $L_{23 i+1}, L_{23 i+2}, \ldots, L_{23 i+23}$ will list $R_{1}, R_{2}, \ldots, R_{23}$, as well as all liars except for themselves. The 16 liars $L_{185}, L_{186}, \ldots, L_{200}$ will list $R_{1}, R_{2}, \ldots, R_{23}, L_{1}, L_{2}, \ldots, L_{177}$ in their lists. The tourist, upon seeing the received lists, will not be able to reliably identify any liar among $L_{1}, L_{2}, \ldots, L_{184}$ (which means he will identify no more than 16 liars). Indeed, if the tourist were to claim that $L_{k}$ is a liar, he could be wrong: the group of 23 people, which included $L_{k}$, could consist entirely of knights who, in their lists, uniformly indicated all other people who could turn out to be liars.

Now, let's understand why the tourist will always be able to identify at least 16 liars. The fact that a person lists 200 names of supposed liars is equivalent to them calling the remaining 23 people (including themselves) knights. All 23 knights will definitely call each other knights. Note that if a person $A$ calls a group of 23 people knights, and not all of them call the same group of people knights, then $A$ is definitely a liar (if he were a knight, then all the people he named would also be knights, and they would call the same group of people knights). There are no more than 9 groups of 23 people who call each other knights (since $10 \cdot 23 > 223$), and these groups do not overlap. Therefore, the tourist can definitely guarantee that the remaining people, who do not belong to such groups, are liars. And there are at least $223 - 9 \cdot 23 = 16$ such people.