A sunflower anti-Ramsey theorem and its applications

A $h$-sunflower in a hypergraph is a family of edges with $h$ vertices in common. We show that if we colour the edges of a complete hypergraph in such a way that any monochromatic $h$-sunflower has at most $\lambda$ petals, then it contains a large rainbow complete subhypergraph. This extends a theorem by Lefmann, R\"odl and Wysocka, but this version can be applied to problems in geometry and algebra. We also give an infinite version of the theorem.