= Minimum-degree Ramsey lower bound
For a finite <simple graph> $H$ of <minimum degree of a graph> $d$, its ordinary two-colour <graph Ramsey number> satisfies
$$
R(H)\geq2^{d/2}.
$$
The lower bound also applies when $H$ contains a subgraph of minimum degree at least $d$, since a monochromatic copy of $H$ would contain a monochromatic copy of that subgraph. The paragraph before Corollary1.7 of <Ordered Ramsey numbers by David Conlon and others> records this ordinary Ramsey bound before comparing it with ordered Ramsey numbers.<Ordered Ramsey numbers by David Conlon and others>[Printed p.5, paragraph immediately before Corollary1.7]{ref}
For $H=K_t$, this gives $R(K_t)\geq2^{(t-1)/2}$. Thus even without using the entire degree sequence, a large minimum degree already forces an exponential obstruction to guaranteeing a monochromatic copy.
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