According to the reference 1 below to earlier work of the authors in this paper [here][1], 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][2] 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.


  [1]: https://www.cs.rit.edu/~spr/PUBL/paper44.pdf
  [2]: http://dcs.gla.ac.uk/~alice/papers/ramseyConstraints2016.pdf