Abstract In a recent paper, Brendle showed the uniqueness of Bryant soliton among 3-dimensional κ-solutions. this we present an alternative proof for fact and show that compact κ-solutions are rotationally symmetric. Our arose from independent work relating to our Strong Stability Theorem singular Ricci flows.

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2κ = κ+, another for which 2κ = κ++ ...

For α, β, n ordinals α → (β, n) means that any graph on the ordinal α either contains an independent set of order-type β or a complete subgraph of order-type n. We only consider cases where n is a natural number, hence the naming. The partition relation α → (β, n) for α, β finite multiples of κ where κ = ω or κ is weakly compact is characterized by a finitary problem as is the same for α, β fin...

Let B(κ, λ) be the subalgebra of P(κ) generated by [κ]. It is shown that if B is any homomorphic image of B(κ, λ) then either |B| < 2 or |B| = |B|, moreover if X is the Stone space of B then either |X| ≤ 2 λ or |X| = |B| = |B|. This implies the existence of 0-dimensional compact T2 spaces whose cardinality and weight spectra omit lots of singular cardinals of “small” cofinality.

Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at κ, assuming that κ = κ and there is a weakly compact cardinal above κ. If in addition κ is supercompact then we can force κ to be א...

The subalgebra of the tautological ring of the moduli of curves of compact type generated by the κ classes is studied. Relations, constructed via the virtual geometry of the moduli of stable maps, are used to prove universality results relating the κ rings in genus 0 to higher genus. Predictions for κ classes of the Gorenstein conjecture are proven.

Krishnamoorthy, Thulasiraman and Swamy [Minimum order graphs with specified diameter, connectivity and regularity, Networks 19 (1989) 25–46] showed that a δ-regular graph with diameter D at most 3 has (vertex-)connectivity κ at least 2, and if D ≤ 2 then the connectivity is at least κ ≥ min{δ, 3}. Likewise, Soneoka, Nakada, Imase and Peyrat [Sufficient conditions for maximally connected graphs,...

The minimum weight of a nowhere first-countable compact space of countable π-weight is shown to be κB, the least cardinal κ for which the real line R can be covered by κ many nowhere dense sets.

In the context of large cardinals, the classical diamond principle 3κ is easily strengthened in natural ways. When κ is a measurable cardinal, for example, one might ask that a 3κ sequence anticipate every subset of κ not merely on a stationary set, but on a set of normal measure one. This is equivalent to the existence of a function ` ... κ → Vκ such that for any A ∈ H(κ) there is an embedding...

A space X is truly weakly pseudocompact if X is either weakly pseudocompact or Lindelöf locally compact. We prove: (1) every locally weakly pseudocompact space is truly weakly pseudocompact if it is either a generalized linearly ordered space, or a protometrizable zero-dimensional space with χ(x,X) > ω for every x ∈ X; (2) every locally bounded space is truly weakly pseudocompact; (3) for ω < κ...