Consider Ramsey's theorem:
For any $s, t \ge 1$, there exist $R(s, t) < \infty$ such that any graph on $R(s, t)$ vertices contains either an independent set of size $s$ or a clique of size $t$. In particular,
$$R(s, t) \le \binom{s+t-2}{s-1}$$
Is this an exclusive condition? In other words, can an independent set of size $s$ exist along side a clique of size $t$ in a graph with $R(s,t)$ vertices?