Maximum Semiorders in Interval Orders

Let $s(n)$ be the largest integer such that every n-point interval order includes an $s(n)$-point semiorder. Equivalently, $s(n)$ is the largest integer such that every n-point interval graph includes an $s(n)$-point unit interval graph. Although $s(n - 1) = s(n) = n/2 + 1$ for even n from 4 to 14, this pattern does not persist since $s(17)=9$. In addition, $s(n) > n/\log _2 n$ for $n\geqq 3$, and $s(n)/n \to 0$. It is conjectured that $s(n) (\log _2 n)/n \to c$ for some c in [1, 3].