Let R be a connected 2-manifold without boundary obtained from a (possibly infinite) collection of polygons by identifying them along edges of equal length. Let V be the set of vertices, and for every v ∈ V , let κ(v) denote the (Gaussian) curvature of v: 2π minus the sum of incident polygon angles. Descartes showed that ∑ v∈V κ(v) = 4π whenever R may be realized as the surface of a convex poly...

Configurations of type (κ +1)κ give rise to κ–regular simple graphs via configuration graphs. On the other hand, neighbourhood geometries of C4–free κ–regular simple graphs on κ 2 + 1 vertices turn out to be configurations of type (κ + 1)κ. We investigate which configurations of type (κ +1)κ are equal or isomorphic to the neighbourhood geometry of their configuration graph and conversely. We cl...

We introduce a general method of constructing locally compact scattered spaces from certain families of sets and then, with the help of this method, we prove that if κ = κ then there is such a space of height κ with only κ many isolated points. This implies that there is a locally compact scattered space of height ω2 with ω1 isolated points in ZFC, solving an old problem of the first author.

Our main results are: 1) every countably certified extender that coheres with the core model K is on the extender sequence of K, 2) K computes successors of weakly compact cardinals correctly, 3) every model on the maximal 1-small construction is an iterate of K, 4) (joint with W. J. Mitchell) K‖κ is universal for mice of height ≤ κ whenever κ ≥ א2, 5) if there is a κ such that κ is either a si...

We prove that if there are c incomparable selective ultrafilters then, for every infinite cardinal κ such κω=κ, exists a group topology on the free Abelian of cardinality without nontrivial convergent sequences and finite power is countably compact. In particular, arbitrarily large compact groups. This answers 1992 question D. Dikranjan Shakhmatov.

We prove that for regular λ above a strong limit singular µ certain guessing principles follow just from cardinal arithmetic assumptions. The main result is that for such λ and µ there are coboundedly many regular κ < µ such that ♣ − (S λ κ) holds whenever λ = λ <κ. 1 0 Introduction. The main result of this note is that for any regular cardinal λ above ω there are unboundedly many regular cardi...

Let κ be a regular uncountable cardinal and λ ≥ κ. The principle of stationary reflection for Pκλ has been successful in settling problems of infinite combinatorics in the case κ = ω1. For a greater κ the principle is known to fail at some λ. This note shows that it fails at every λ if κ is the successor of a regular uncountable cardinal or κ is countably closed.

We prove that for κ an uncountable cardinal, there exist 2κ many non homeomorphic weakly compact convex subsets of weight κ in the Hilbert space l2(κ).

Juhasz and van Mill denote by dis(X) the least cardinal of a cover of X by discrete subspaces. They show that dis(X) ≥ c (and hence the answer to Question 1.1 is positive) for any compact crowded hereditarily normal X. In fact, this follows from their stronger result that for such X, rs(X) + ls(X) ≥ c, where rs(X) (resp., ls(X)) is the least cardinal of a cover of X by right(resp., left)-separa...