Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 161 / 4 / iv / Solution

Put $k=p^{3p}$. Parts ii and iii give a graph with neither a clique nor an independent set of size $k$. Its number of vertices satisfies the standard binomial lower boundConsequently the modular-intersection graph Ramsey lower bound gives

For every fixed $C>0$,The first quantity exceeds the second when $p>3C$. Thus $p^{p^2}$ is eventually larger than $k^C$ for every fixed $C$, so this lower bound grows faster than every polynomial in $k$.