Let G be a graph with n vertices and independence number α. Hadwiger’s conjecture implies that G contains a clique minor of order at least n/α. In 1982, Duchet and Meyniel proved that this bound holds within a factor 2. Our main result gives the first improvement on their bound by an absolute constant factor. We show that G contains a clique minor of order larger than .504n/α. We also prove rel...

We present a technique for coding sets “into K,” where K is the core model below a strong cardinal. Specifically, we show that if there is no inner model with a strong cardinal then any X ⊂ ω1 can be made ∆3 (in the codes) in a reasonable and stationary preserving set generic extension.

We use model theoretic forcing to study and generalize the construction of (K ,≤)-generic models introduced by Kueker and Laskowski. We characterize the (K ,≤)-generic models in terms of forcing and introduce a more general class of models, called essential forcing generics, which have many of the same properties.

We introduce and study the first-order Generic Vopěnka’s Principle, which states that for every definable proper class of structures C of the same type, there exist B 6= A in C such that B elementarily embeds into A in some set-forcing extension. We show that, for n ≥ 1, the Generic Vopěnka’s Principle fragment for Πn-definable classes is equiconsistent with a proper class of n-remarkable cardi...

We present some results about generics for computable Mathias forcing. The n-generics and weak n-generics in this setting form a strict hierarchy as in the case of Cohen forcing. We analyze the complexity of the Mathias forcing relation, and show that if G is any n-generic with n ≥ 3 then it satisfies the jump property G(n−1) = G′ ⊕ ∅(n). We prove that every such G has generalized high degree, ...

Various theorems for the preservation of set-theoretic axioms under forcing are proved, regarding both forcing axioms and axioms true in the Levy-Collapse. These show in particular that certain applications of forcing axioms require to add generic countable sequences high up in the set-theoretic hierarchy even before collapsing everything down to א1. Later we give applications, among them the c...

Zero forcing can be described as a graph process that uses color change rule in which vertices white to blue. The throttling number of minimizes the sum initially colored blue and time steps required entire graph. Positive semidefinite (PSD) zero is commonly studied variant standard alters rule. This paper introduces method for extending using PSD process. Using this extension method, graphs wi...

We look for a parallel to the notion of " proper forcing " among λ-complete forcing notions not collapsing λ +. We suggest such a definition and prove that it is preserved by suitable iterations. This work follows [Sh 587] and [Sh 667] (and see history there), but we do not rely on those papers. Our goal in this and the previous papers is to develop a theory parallel to " properness in CS itera...

This article continues Ros lanowski and Shelah [8, 9, 10, 11, 12] and we introduce here a new property of (<λ)–strategically complete forcing notions which implies that their λ–support iterations do not collapse λ (for a strongly inaccessible cardinal λ).

Let convergences λi : B → P (B), i ≤ 4, on a complete Boolean algebra B be defined in the following way. For a sequence x = 〈xn : n ∈ ω〉 in B and the corresponding B-name for a subset of ω, τx = {〈ň, xn〉 : n ∈ ω}, let λi(x) = { {‖τx is infinite‖} if bi(x) = 1B, ∅ otherwise, where b1(x) = ‖τx is finite or cofinite‖, b2(x) = ‖τx is not unsupported‖, b3(x) = ‖τx is not a splitting real‖ and b4(x) ...