Solution:

We must assume that nobody was eaten by more than one person for the problem statement to make sense. To each cannibal we assign a numerical "depth" as follows: the depth of cannibal $C$ is the largest integer $n$ such that there exist cannibals $C_{1}, C_{2}, \ldots, C_{n}=C$ such that $C_{i}$ ate $C_{i+1}$ for $i=1,2, \ldots, n-1$. (A cannibal who was eaten by no one has depth 1. Also note that depth is definable: there is an upper bound on the value of $n$, since any chain of length greater than 30 would contain some cannibal twice, an impossibility; hence there is a maximum value of $n$ for which chains exist.) Note that no cannibal ate another of the same depth, since the inner cannibal always has higher depth.

Now, if we can find a cannibal of depth $\geq 6$ we are done, so assume that the only depths which occur are $1,2,3,4,5$. By the pigeonhole principle, some depth was assumed by at least 6 cannibals; from the given, one of these six ate another. But we know this is impossible, so our assumption was wrong and some higher depth does occur.