Saturated and weakly saturated hypergraphs

Weakly Saturated Hypergraphs and Exterior Algebra

On hamiltonian chain saturated uniform hypergraphs

We say that a hypergraph H is hamiltonian chain saturated if H does not contain a hamiltonian chain but by adding any new edge we create a hamiltonian chain in H. In this paper we ask about the smallest size of a k-uniform hamiltonian chain saturated hypergraph. We present a construction of a family of k-uniform hamiltonian chain saturated hypergraphs with O(nk−1/2) edges.

Hamilton-chain saturated hypergraphs

We say that a hypergraph H is hamiltonian path (cycle) saturated if H does not contain an open (closed) hamiltonian chain but by adding any new edge we create an open (closed) hamiltonian chain in H. In this paper we ask about the smallest size of an r-uniform hamiltonian path (cycle) saturated hypergraph, mainly for r = 3. We present a construction of a family of 3-uniform path (cycle) saturat...

Size of weakly saturated graphs

Let P be a hereditary property. Let kP(G) denote the number of forbidden subgraphs, which are contained in G. A graph G is said to be weakly P-saturated, if G ∈ P and the edges of the complement of G can be labelled e1, e2, . . . , el in such way that for i= 0, 1, . . . , l− 1 the inequality kP(Gi+1)> kP(Gi) holds, whereG0=G,Gi+1=Gi + ei andGl =Kn. The minimum possible size of weakly P-saturate...

Less Saturated Ideals

We prove the following: (1) If κ is weakly inaccessible then NSκ is not κ+-saturated. (2) If κ is weakly inaccessible and θ < κ is regular then NSθ κ is not κ +saturated. (3) If κ is singular then NS κ+ is not κ++-saturated. Combining this with previous results of Shelah, one obtains the following: (A) If κ > א1 then NSκ is not κ+-saturated. (B) If θ+ < κ then NSθ κ is not κ +-saturated.