We show that countable increasing unions preserve a large family of well-studied covering properties, which are not necessarily σ-additive. Using this, together with infinite-combinatorial methods and simple forcing theoretic methods, we explain several phenomena, settle problems of Just, Miller, Scheepers and Szeptycki [15], Gruenhage and Szeptycki [13], Tsaban and Zdomskyy [33], and Tsaban [2...

We give characterizations for the (in ZFC unprovable) sentences “Every Σ12–set is measurable” and “Every ∆ 1 2–set is measurable” for various notions of measurability derived from well–known forcing partial orderings.

We consider vertex colorings of graphs in which adjacent vertices have distinct colors. A graph is s-chromatic if it is colorable in s colors and any coloring of it uses at least s colors. The forcing chromatic number Fχ(G) of an s-chromatic graph G is the smallest number of vertices which must be colored so that, with the restriction that s colors are used, every remaining vertex has its color...

For two vertices u and v in a graph G = (V, E), the detour distance D (u, v) is the length of a longest u – v path in G. A u – v path of length D (u, v) is called a u – v detour. For subsets A and B of V, the detour distance D (A, B) is defined as D (A, B) = min {D (x, y) : x ∈ A, y ∈ B}. A u – v path of length D (A, B) is called an A – B detour joining the sets A, B V where u ∈ A and v ∈ B. A...

We consider vertex colorings of graphs in which adjacent vertices have distinct colors. A graph is s-chromatic if it is colorable in s colors and any coloring of it uses at least s colors. The forcing chromatic number Fχ(G) of an s-chromatic graph G is the smallest number of vertices which must be colored so that, with the restriction that s colors are used, every remaining vertex has its color...

The zero forcing number is a graph invariant introduced in order to study the minimum rank of the graph. In the first part of this paper, we first highlight that the computation of the zero forcing number of any directed graph (allowing loops) is NP-hard. Furthermore, we identify a class of directed trees for which the zero forcing number is computable in linear time. The second part of the pap...

We show that א2 ≤ b < g is consistent. This work is dedicated to James Baumgartner on the occasion of his

The zero forcing number is a graph invariant introduced in order to study the minimum rank of the graph. In the first part of this paper, we first highlight that the computation of the zero forcing number of any directed graph (allowing loops) is NP-hard. Furthermore, we identify a class of directed trees for which the zero forcing number is computable in linear time. The second part of the pap...

We prove that two basic questions on outer measure are undecid-able. First we show that consistently • every sup-measurable function f : R 2 −→ R is measurable. The interest in sup-measurable functions comes from differential equations and the question for which functions f : R 2 −→ R the Cauchy problem y ′ = f (x, y), y(x 0) = y 0 has a unique almost-everywhere solution in the class AC l (R) o...

The class forcing theorem, which asserts that every class forcing notion P admits a forcing relation P, that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel-Bernays set theory GBC to the principle of elementary transfinite recursion ETROrd for class...