Classical Helly's theorem states that if n>d subsets of d-dimensional Euclidean space are such that each d+1 of them have nonempty intersection then the intersection of all is non empty. This theorem can be seen as a theorem on intersection patterns of convex sets.

The question of what are possible intersection patterns of convex sets can be formalized via simplicial complexes. This question is algorithmically hard; however, there are many necessary (and also sufficient) conditions.

The task of the talk is to introduce this topic and also present some new results such as nonrepresentability of finite projective planes by convex sets in a fixed dimension.