On the number of independent orders

We investigate a model theoretic invariant ${\kappa }_{srd}^m(T)$, which was introduced by Shelah in his famous book, and prove that ${\kappa }_{srd}^m(T)$ is sub-additive. When ${\kappa }_{srd}^m(T)$ is infinite, this gives the equality ${\kappa }_{srd}^m(T)={\kappa }_{srd}^1(T)$, answering a question by Shelah. We apply the same proof method to analyze another invariant ${\kappa }_{ird}^m(T)$, and show that it is also sub-additive, improving a result in the book.