We study properties of Σ1 and Π 1 1 subsets of ω ω that are cofinal relative to the orders ≤ (≤?) of full (eventual) domination. We apply these results to prove that the topological statement “Any compact covering mapping from a Borel space onto a Polish space is inductively perfect” is equivalent to the statement “∀α ∈ ω , ω ∩ L(α) is bounded for ≤?”. This work is a continuation of [3], in whi...

We prove that the topological statement: “Any compact covering mapping between two Borel sets is inductively perfect” is equivalent to the set-theoretical statement: “ ∀α ∈ ωω , א 1 < א1”. The starting point of this work is the following topological problem: Problem. Is any compact covering mapping between two Borel spaces inductively perfect? We recall that if f : X → Y is a continuous and ont...

Roussel and Rubio proved a lemma which is essential in the proof of the Strong Perfect Graph Theorem. We give a new short proof of the main case of this lemma. In this note, we also give a short proof of Hayward’s decomposition theorem for weakly chordal graphs, relying on a Roussel–Rubio-type lemma. We recall how Roussel–Rubio-type lemmas yield very short proofs of the existence of even pairs ...

We will extend Reed's Semi-Strong Perfect Graph Theorem by proving that unbreakable C 5-free graphs diierent from a C 6 and its complement have unique P 4-structure.

Intrusion detection systems are designed to provide security in computer networks, so that if the attacker crosses other security devices, they can detect and prevent the attack process. One of the most essential challenges in designing these systems is the so called curse of dimensionality. Therefore, in order to obtain satisfactory performance in these systems we have to take advantage of app...

A set X ⊆ 2 has property (s) (Marczewski (Szpilrajn)) iff for every perfect set P ⊆ 2 there exists a perfect set Q ⊆ P such that Q ⊆ X or Q∩X = ∅. Suppose U is a nonprincipal ultrafilter on ω. It is not difficult to see that if U is preserved by Sacks forcing, i.e., it generates an ultrafilter in the generic extension after forcing with the partial order of perfect sets, then U has property (s)...

Lam, Gusfield, and Sridhar (2009) showed that a set of three-state characters has a perfect phylogeny if and only if every subset of three characters has a perfect phylogeny. They also gave a complete characterization of the sets of three three-state characters that do not have a perfect phylogeny. However, it is not clear from their characterization how to find a subset of three characters tha...