Fano Kaleidoscopes and their generalizations

In this work we introduce Fano Kaleidoscopes, Hesse Kaleidoscopes and their generalizations. These are a particular kind of colored designs for which we will discuss general theory, present some constructions and prove existence results. In particular, using difference methods we show the existence of both a Fano and a Hesse Kaleidoscope on v points when v is a prime or prime power congruent to 1(mod6), v≠13. In the Fano case this, together with known results on pairwise balanced designs, allows us to prove the existence of Kaleidoscopes of order v for many other values of v; we discuss what the situation is, on the other hand, in the Hesse and general case.