Decompositions of Hypergraphs into Delta-systems and Constellations

Abstract

A partition of the edge set of a hypergraph $H$ into subsets inducing hypergraphs $H_1,\dots ,H_r$ is said to be a $decomposition$ of $H$ into $H_1,\dots ,H_r$. A uniform hypergraph $F=(\bigcup \mathcal{F},\mathcal{F})$ is a $\mathrm{\Delta }$-$system$ if there is a set $K\subseteq V(F)$, called the $kernel$ of $F$, such that $A\cap B=K$ for every $A,B\in \mathcal{F}$, $A\ne B$. A disjoint union of $\mathrm{\Delta }$-systems whose kernels have the same cardinality is said to be a $constellation$. In the paper, we find sufficient conditions for the existence of a decomposition of a hypergraph $H$ into:
a) $\mathrm{\Delta }$-systems having almost equal sizes and kernels of the same cardinality,
b) isomorphic copies of constellations such that the sizes of their components are relatively prime.

In both cases, the sufficient conditions are satisfied by a wide class of hypergraphs $H$.