According to the reference 1 below to earlier work of the authors in this paper here, it is known that $$ 30\leq R(3,3,4)\leq 31 $$

1. Piwakowski, K., and Radziszowski, S. P., Journal of Combinatorial Mathematics and Combinatorial Computing, 27:135-141, 1998.

According to the reference 1 below to earlier work of the authors in this paper here, it is known that $$ 30\leq R(3,3,4)\leq 31 $$

Edit: A more recent paper, by Codish, Frank, Itzhakov and Miller, available here has shown that $R(3,3,4)=30.$ Thanks to @Julian.

1. Piwakowski, K., and Radziszowski, S. P., Journal of Combinatorial Mathematics and Combinatorial Computing, 27:135-141, 1998.