  Finding exact Ramsey numbers is a problem typically restricted to relatively
small graphs. The flag algebra method was developed to find asymptotic results
for very large graphs, so it seems that the method is not suitable for finding
small Ramsey numbers. But this intuition is wrong, and we will develop a
technique to do just that in this paper.
  We find new upper bounds for many small graph and hypergraph Ramsey numbers.
As a result, we prove the exact values $R(K_4^-,K_4^-,K_4^-)=28$, $R(K_8,C_5)=
29$, $R(K_9,C_6)= 41$, $R(Q_3,Q_3)=13$, $R(K_{3,5},K_{1,6})=17$, $R(C_3, C_5,
C_5)= 17$, and $R(K_4^-,K_5^-;3)= 12$.
  We hope that this technique will be adapted to address other questions for
smaller graphs with the flag algebra method.
