Note: this isn't a HW question. I'm doing problems from the book "Invitation to Discrete Mathematics- Jirí Matousek, Jaroslav Nesetril"
The following is the question:
I tried by contradiction but couldn't make any progress.
Any help is highly appreciated.
Since $X$ is finite, so is the set of its possible relations, $P(X\times X)$ (with cardinality $2^{|X|^2}=:N$), so any sequence of at least $N+1$ relations over $X$ must contain a repetition.
So does in particular the infinite sequence $R, R^2, R^3, \dots$.