1. Introduction Let V [G] be the result of adding a Cohen real and a Cohen subset of ω 2 to a model V of the GCH. Every infinite cardinal gets a new subset in V [G], namely, the Cohen real. Yet, in some sense, only ω and ω 2 get new subsets. One way to capture this is to say that a sort of dual to Covering holds between V and V [G], namely, " κ-cocovering " for κ other than ω and ω 2. We need s...

A simple digraph describes the off-diagonal zero-nonzero pattern of a family of (not necessarily symmetric) matrices. Minimum rank of a simple digraph is the minimum rank of this family of matrices; maximum nullity is defined analogously. The simple digraph zero forcing number is an upper bound for maximum nullity. Cut-vertex reduction formulas for minimum rank and zero forcing number for simpl...

The zero forcing number Z(G) is used to study the minimum rank/maximum nullity of the family of symmetric matrices described by a simple, undirected graph G. The positive semidefinite zero forcing number is a variant of the (standard) zero forcing number, which uses the same definition except with a different color-change rule. The positive semidefinite maximum nullity and zero forcing number f...

Zero forcing is a deterministic iterative graph coloring process in which vertices are colored either blue or white, and every round, any that have single white neighbor force these to become blue. Here we study probabilistic zero forcing, where non-zero probability of each We explore the propagation time for on Erd?s–Réyni random G ( n , p ) when start with vertex show = log ? o 1 then high it...

With every σ-ideal I on a Polish space we associate the σ-ideal generated by closed sets in I. We study the forcing notions of Borel sets modulo the respective σ-ideals and find connections between their forcing properties. To this end, we associate to a σ-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the...