In this text, we begin by giving a definition of Vector Space Ramsey Numbers. It concerns colourings of $t$-dimensional subspaces of some vector space $\mathbb{F}_q^n$. We want to ensure that each colouring contains a monochromatic $k$-dimensional subspace. After proving that these numbers always exist, we continue with studying asymptotic bounds for these numbers. We study a selection of methods, such as through coding theory or using the probabilistic method, to obtain lower and upper bounds for vector space Ramsey numbers. Lastly, we introduce two methods to directly compute vector space Ramsey numbers. That being through an ILP formulation and through a SAT formulation.