Does there exist a graph $G$ which contains no $K_4$, and yet any $2$-colouring of the edges produces a monochromatic $K_3$?

Does there exist a graph $G$ which contains no $K_4$, and yet any $2$-colouring of the edges produces a monochromatic $K_3$?

Erdős and Hajnal [ErHa67] first asked for the existence of any such graph. Existence was proved by Folkman [Fo70], but with very poor quantitative bounds. (As a result these quantities are often called Folkman numbers.) The current best bounds on the minimal number of vertices $N$ of such a graph are $21 \leq N \leq 786$, where the lower bound is due to Bikov and Nenov [BiNe20] and the upper bound is due to Lange, Radziszowski, and Xu [LRX14].