Binomial upper bound for a Ramsey number
= Binomial upper bound for a Ramsey number
The off-diagonal recursion
$$
R(s,t)\leq R(s-1,t)+R(s,t-1)
$$
with $R(1,t)=R(s,1)=1$ gives
$$
R(s,t)\leq\binom{s+t-2}{s-1}.
$$
In particular, $R(t)\leq\binom{2t-2}{t-1}<2^{2t}$.