Lesson 6

Problem. Let \((X,\preceq)\) be a poset and \(|X|=5\). Consider the set of (ordered) pairs of comparable elements: \[S=\{(x,y) \in X^{2} \colon x \preceq y\} \ .\] It is easy to see that \(|S| \ge 5\): \((x,x) \in S\) for any \(x \in X\) (by reflexivity). Find the maximum possible size of \(S\).