Erdős Problem #6 — Monotone runs among three consecutive prime gaps

Let $2 = p_1 < p_2 < \cdots$ be the sequence of consecutive primes. [$p_{n+1} - p_n = d_n$.] Turán and I [15] proved that the inequalities $d_{n+1} > d_n$ and $d_{n+1} < d_n$ both have infinitely many solutions. We have not been able to prove that $d_n > d_{n+1} > d_{n+2}$ or $d_n < d_{n+1} < d_{n+2}$ have infinitely many solutions. In fact we cannot disprove the existence of an integer $n_0$ so that for every $k \geq 0$, $d_{n_0}+2k > d_{n_0}+2k+1$ and $d_{n_0}+2k+1 < d_{n_0}+2k+2$.