We introduce the study of forcing sets in mathematical origami. The origami material folds flat along straight line segments called creases, each of which is assigned a folding direction of mountain or valley. A subset F of creases is forcing if the global folding mountain/valley assignment can be deduced from its restriction to F . In this paper we focus on one particular class of foldable pat...

For an ordered set W = {w1, w2, . . . , wk} of vertices and a vertex v in a connected graph G, the (metric) representation of v with respect to W is the k-vector r(v|W ) = (d(v,w1), d(v, w2), . . . , d(v, wk)), where d(x, y) represents the distance between the vertices x and y. The setW is a resolving set for G if distinct vertices of G have distinct representations. A resolving set of minimum ...

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 prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler’s classical result in the theory of forcing, and the stat...

We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler’s classical result in the theory of forcing, and the stat...

We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary sub-models of some (H(χ),∈). This leads to forcing notions which are “reasonably” definable. We present two specific properties materializing this intuition: nep (non-elementary properness) and snep (Souslin non-elementary properness) and al...

We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler’s classical result in the theory of forcing, and the stat...

Let G be a graph that admits a perfect matching. A forcing set for a perfect matching M of G is a subset S of M , such that S is contained in no other perfect matching of G. This notion originally arose in chemistry in the study of molecular resonance structures. Similar concepts have been studied for block designs and graph colorings under the name defining set, and for Latin squares under the...

Zero forcing (also called graph infection) on a simple, undirected graph G is based on the colorchange rule: If each vertex of G is colored either white or black, and vertex v is a black vertex with only one white neighbor w, then change the color of w to black. A minimum zero forcing set is a set of black vertices of minimum cardinality that can color the entire graph black using the color cha...

We investigate the effect of adding a single real (for various forcing notions adding reals) on cardinal invariants associated with the continuum. We show: (1) adding an eventually different or a localization real adjoins a Luzin set of size continuum and a mad family of size ω1; (2) Laver and Mathias forcing collapse the dominating number to ω1, and thus two Laver or Mathias reals added iterat...