Erdős Problem #22 — $K_4$-free graphs with $n^2/8$ edges and $o(n)$ independence number

There remains the following problem. Does there exist a $G(n, [n^2/8])$ without a $K_4$ and at most $o(n)$ independent points? (As usual, $G(n, m)$ denotes a graph with $n$ points and $m$ edges.) At present we do not see a promising line of attack. The most we could hope for is the following. For every $\eta > 0$ there exists an $\epsilon > 0$ such that whenever $n$ is sufficiently large, some $G = G(n, [(n^2/8)(1 + \epsilon)])$ satisfies $I(G) < \eta n$ and $\alpha(G) < 4$.