Decomposable cardinals.

Constructing vertex decomposable graphs

‎Recently‎, ‎some techniques such as adding whiskers and attaching graphs to vertices of a given graph‎, ‎have been proposed for constructing a new vertex decomposable graph‎. ‎In this paper‎, ‎we present a new method for constructing vertex decomposable graphs‎. ‎Then we use this construction to generalize the result due to Cook and Nagel‎.

Remarkable cardinals

To a large extent, the scientific work of both Peter Koepke and Philip Welch has always been inspired by Ronald Jensen’s fine structure theory and his Covering Lemma for L, the constructible universe. Cf., e.g., their joint papers [5] and [6]. In this paper we aim to play with the theme that large cardinals compatible with “V=L,” specifically: remarkable cardinals, allow us to create situations...

Ultrahuge cardinals

Jónsson Cardinals, Erdös Cardinals, and The Core Model

We show that if there is no inner model with a Woodin cardinal and the Steel core model K exists, then every Jónsson cardinal is Ramsey in K, and every δ-Jónsson cardinal is δ-Erdős in K. In the absence of the Steel core model K we prove the same conclusion for any model L[E ] such that either V = L[E ] is the minimal model for a Woodin cardinal, or there is no inner model with a Woodin cardina...

Fallen cardinals

We prove that for every singular cardinal μ of cofinality ω, the complete Boolean algebra compPμ(μ) contains a complete subalgebra which is isomorphic to the collapse algebra CompCol(ω1, μ0 ). Consequently, adding a generic filter to the quotient algebra Pμ(μ) = P(μ)/[μ] collapses μ0 to א1. Another corollary is that the Baire number of the space U(μ) of all uniform ultrafilters over μ is equal ...