Decomposition of large combinatorial structures (Q1107544)

Let H be a hypergraph and let F be a family of hypergraphs. A partition of the edge set of H is called an F-decomposition if each set from the partition induces in H a hypergraph isomorphic to a member of F. Let \(F_{n,k}(p,q)\) be an n-uniform hypergraph with vertex set X and edge set \(E=\{e_ 1,...,e_ k\}\) satisfying the conditions: (1) there is \(P\subseteq X,| P| =p\), such that for every \(2\leq i<j\leq k\) \(e_ i\cap e_ j=P\), (2) there is \(Q\subseteq P,| Q| =q\), such that for every \(2\leq i\leq k\) \(e_ i\cap e_ 1=Q\). Let \(\Pi_{n,k,r}=\{F_{n,k}(p,q):\) \(0\leq q\leq p\leq r\leq n-1\}.\) We prove the following theorem: Let n,k,r be given integers, \(0\leq r\leq n-1\). There exist a positive constant \(c=c(n,k,r)\) and a minimal integer C(n,k,r) such that every n- uniform hypergraph H with e(H)\(\geq C(n,k,r)\) and \(\Delta\) (H)\(\leq ce(H)\) has a \(\Pi_{n,k,r}\)-decomposition. \((e(H)=| E(H)|\) and \(\Delta\) (H) is the maximal degree of a vertex in H.) This result improves and generalizes earlier results of Alon, Caro, and Lonc and Truszczynski. Generalizations to other combinatorial structures are given.