Density of Ramsey Graphs Suppose we have a graph $G$ with $n$ vertices that contains neither a clique of size $3 \log(n)$ nor an independent set of size $3 \log(n)$ (for example $G(n,0.5)$ satisfies this property with with high probability). Is it true that the number of edges of $G$ is at least $n^2/100$, i.e., it cannot be too sparse?

More generally, I would like to know whether such graphs have some kind of pseudo-random properties.
The answer to your question is yes. It is a result of Erdős and Szemerédi from 1972. We know a little, but non-zero about pseudo-random properties of Ramsey graphs. For example, we know that many different kinds of induced subgraphs appear in such a graph. See, for example, a paper by Alon–Balogh–Kostochka–Samotij and references therein.