(a) Define the Ramsey numbers $R(s, t)$ and $R(s)$ for integers $s, t \geqslant 2$. Show that $R(s, t)$ exists for all $s, t \geqslant 2$ and that if $s, t \geqslant 3$ then $R(s, t) \leqslant R(s-1, t)+R(s, t-1)$.

(b) Show that, as $s \rightarrow \infty$, we have $R(s)=O\left(4^{s}\right)$ and $R(s)=\Omega\left(2^{s / 2}\right)$.

(c) Show that, as $t \rightarrow \infty$, we have $R(3, t)=O\left(t^{2}\right)$ and $R(3, t)=\Omega\left(\left(\frac{t}{\log t}\right)^{3 / 2}\right)$.

[Hint: For the lower bound in (c), you may wish to begin by modifying a random graph to show that for all $n$ and $p$ we have

$$R(3, t)>n-\left(\begin{array}{l} n \\ 3 \end{array}\right) p^{3}-\left(\begin{array}{c} n \\ t \end{array}\right)(1-p)\left(\begin{array}{c} t \\ 2 \end{array}\right)$$