Ramsey numbers are mostly unknown,I have seen bounds on Ramsey numbers ( for example via the probabilistic method) but there is more information about Turán numbers, , and i can't find in the internet a bound on the diagonal Ramsey numbers based on known Turan numbers.
Here is why i am suspicious that such a relation must exist : If we consider 2-coloring of the complete graph $K_n$ and we regard (say) only red edges, then if there is more than $t_r(n)$ red edges then there must be a red clique of order $r$,and on the other hand if there is this much blue edges we have a blue clique of size $r$,so a bound must exist which relates Ramsey number $R(r,r)$ to the Turan number $t_r(n)$.
Please help if you know such a relation (I couldn't find such a result in the older posts). I think something like : smallest number $n$ such that $${n \choose 2}- t_r(n) > t_r(n)$$