Theorem 3 of [this paper][1] by Csikvari, Gyarmati, and Sarkozy shows that $n_0(16)$ does not exist. 

[These slides][2] contain related results and several references. 


  [1]: http://math.mit.edu/~csikvari/ramsey_density.pdf
  [2]: https://www.math.uci.edu/~isaac/Honolulu_LuperiBaglini.pdf