Fano Kaleidoscopes and their generalizations

TL;DR

This paper introduces Fano and Hesse Kaleidoscopes, explores their theoretical foundations, and proves their existence for many prime power orders using difference methods, expanding the understanding of colored combinatorial designs.

Contribution

It defines new classes of colored designs called Fano and Hesse Kaleidoscopes, and establishes their existence for various orders using difference methods and combinatorial constructions.

Findings

Existence of Fano Kaleidoscopes for prime power orders v ≡ 1 mod 6, v ≠ 13.

Existence of Hesse Kaleidoscopes under certain conditions.

Extension of Kaleidoscope existence results to many new orders.

Abstract

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$\pmod{6}$, $v\ne13$. 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.