  The graph $G$ is called a $(3, 3)$-Ramsey graph if in every coloring of the
edges of $G$ in two colors there is a monochromatic triangle. The minimum
number of vertices of the $(3, 3)$-Ramsey graphs without 4-cliques is denoted
by $F_e(3, 3; 4)$. The number $F_e(3, 3; 4)$ is referred to as the most wanted
Folkman number. It is known that $20 \leq F_e(3, 3; 4) \leq 786$.
  In this paper we prove that if $G$ is an $n$-vertex $(3, 3)$-Ramsey graph
without 4-cliques, then $\alpha(G) \leq n - 16$, where $\alpha(G)$ denotes the
independence number of $G$. Using the newly obtained bound on $\alpha(G)$ and
complex computer calculations we obtain the new lower bound $$F_e(3, 3; 4) \geq
21.$$
