Strong compactness and a partition property

Strong Compactness and a Partition Property

We show that if Part(κ, λ) holds for every λ ≥ κ, then κ is strongly compact. Let κ be a regular infinite cardinal, and let λ ≥ κ be a cardinal. Pκ(λ) denotes the set of all subsets of λ of size less than κ. Part(κ, λ) means that for every F : Pκ(λ) × Pκ(λ) → 2, there is a cofinal subset A of (Pκ(λ),⊆) such that F is constant on the set {(a, b) ∈ A × A : a ⊂ b}. This definition is due to Jech [...

A Finite-Compactness Notion, and Property Testing

We investigate properties Π of graphs G of size N such that possession of Π by all f(N)-node subgraphs implies possession of Π by some g(N)-node subgraph. Bounds are given on f(N) and g(N) for bipartiteness. This is compared to a situation in property testing, where edge-induced subgraphs are involved and different, intuitively “better,” bounds are obtained. Bounds are discussed for other prope...

A note on strong compactness and resurrectibility

We construct a model containing a proper class of strongly compact cardinals in which no strongly compact cardinal κ is κ supercompact and in which every strongly compact cardinal has its strong compactness resurrectible.

Identity crises and strong compactness

A Note on Indestructibility and Strong Compactness

If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ+,∞)-distributive and λ is 2λ supercompact, then by [3, Theorem 5], {δ < κ | δ is δ+ strongly compact yet δ isn’t δ+ supercompact} must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in whic...