A graph with independence number alpha is called (alpha,k)-balanced if every induced subgraph on k vertices has independence number alpha as well. We will discuss the maximum number of vertices in an (alpha,k)-balanced graph for fixed k and alpha, a problem with obvious connections to Ramsey Theory. We focus specifically on the case k=2alpha which is motivated from polyhedral combinatorics. (Joint work with A. Brieden, Z. Furedi and R. Ramamurthi)