Turan-type problems in graph theory ask how many edges a graph may have if a certain subgraph is forbidden. One can think of this as an optimization problem, as one is maximizing the global condition of number of edges subject to the local constraint that there is no forbidden subgraph. Problems in combinatorial number theory ask one to deduce properties of a set of (for example) integers while knowing only how large the set is. We study the connection between these two seemingly disjoint areas. Graphs coming from finite projective planes are intimately related to both areas.