The paper is concerned with the computation of covering numbers in the presence of large cardinals. In particular, we revisit Solovay's result that the Singular Cardinal Hypothesis holds above a strongly compact cardinal.

We prove that if there is a model of set-theory which contains no first countable, locally compact, scattered, countably paracompact space $X$, whose Tychonoff square is a Dowker space, then there is an inner model which contains a measurable cardinal.

We construct two models for the level by level equivalence between strong compactness and supercompactness in which if κ is λ supercompact and λ ≥ κ is regular, we are able to determine exactly the number of normal measures $P_{\kappa }\left(\lambda \right)$ carries. In the first of these models, $P_{\kappa }\left(\lambda \right)$ carries $2^{2^{{\left[\lambda \right]}^{<\kappa }}}$ many normal measures, the maximal number. In the second of these models, $P_{\kappa }\left(\lambda \right)$ carries $2^{2^{{\left[\lambda \right]}^{<\kappa }}}$ many normal measures, except if κ is a measurable cardinal which is not a limit of measurable cardinals. In this case, κ (and hence also $P_{\kappa }\left(\kappa \right)$)...

We construct three models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds. In the first two models, below the supercompact cardinal κ, there is a non-supercompact strongly compact cardinal. In the last model, any suitably defined ground model Easton function is realized.

I prove that the statement that “every linear order of size $2^{\omega }$ can be embedded in $({\omega }^{\omega },\ll )$” is consistent with MA + ¬ wKH.

We force and construct a model in which GCH and level by level equivalence between strong compactness and supercompactness hold, along with certain additional “L-like” combinatorial principles. In particular, this model satisfies the following properties: (1) ${♢}_{\delta }$ holds for every successor and Mahlo cardinal δ. (2) There is a stationary subset S of the least supercompact cardinal κ₀ such that for every δ ∈ S, ${◻}_{\delta }$ holds and δ carries a gap 1 morass. (3) A weak version of ${◻}_{\delta }$ holds for every infinite cardinal...

We work towards establishing that if it is consistent that there is a supercompact cardinal then it is consistent that every locally compact perfectly normal space is paracompact. At a crucial step we use some still unpublished results announced by Todorcevic. Modulo this and the large cardinal, this answers a question of S. Watson. Modulo these same unpublished results, we also show that if it is consistent that there is a supercompact cardinal, it is consistent that every locally compact space...