Very many clones above the unary clone

Let ${\mathfrak{c} := 2^{\aleph_0}}$ . We give a family of pairwise incomparable clones on ${{\mathbb N}}$ with ${2^{\mathfrak{c}}}$ members, all with the same unary fragment, namely the set of all unary operations. We also give, for each n, a family of ${2^{\mathfrak{c}}}$ clones all with the same n-ary fragment, and all containing the set of all unary operations.     

PCF arithmetic without and with choice

We deal with relatives of GCH which are provable. In particular, we deal with rank version of the revised GCH. Our motivation was to find such results when only weak versions of the axiom of choice are assumed, but some of the results give us additional information even in ZFC. We also start to deal with pcf for pseudo-cofinality (in ZFC with little choice).     

Universality among graphs omitting a complete bipartite graph

For cardinals λ,κ,θ we consider the class of graphs of cardinality λ which has no subgraph which is (κ,θ)-complete bipartite graph. The question is whether in such a class there is a universal one under (weak) embedding. We solve this problem completely under GCH. Under various assumptions mostly related to cardinal arithmetic we prove non-existence of universals for this problem. We also look at combinatorial properties useful for those problems concerning κ-dense families.     

The spectrum of ultraproducts of finite cardinals for an ultrafilter

We complete the characterization of the possible spectrum of regular ultrafilters D on a set I, where the spectrum is the set of ultraproducts of (finite) cardinals modulo D which are infinite.     

A construction of all normal subgroup lattices of 2-transitive automorphism groups of linearly ordered sets

We give a complete classification and construction of all normal subgroup lattices of 2-transitive automorphism groupsA(Ω) of linearly ordered sets (Ω, ≦). We also show that in each of these normal subgroup lattices, the partially ordered subset of all those elements which are finitely generated as normal subgroups forms a lattice which is closed under even countably-infinite intersections, and we derive several further group-theoretical consequences from our classification. This research was supported by an award from the Minerva-Stiftung, München. The work was done during a stay of the first-named author at The Hebrew University of Jerusalem in fall 1982. He would like to thank his colleagues in Jerusalem for their hospitality and a wonderful time.     

Martin’s axiom does not imply that every two ℕ1-dense sets of reals are isomorphic

Assuming the consistency of ZFC we prove the claim in the title by showing the consistency with ZFC of: There exists a set of realsA such that every function fromA toA is order preserving on an uncountable set. We prove related results among which is the consistency with ZFC of: Every function from the reals into the reals is monotonic on an uncountable set. I would like to thank the United States-Israel Binational Science Foundation for supporting this research by a grant.     

On universal locally finite groups

We deal with the question of existence of a universal object in the category of universal locally finite groups; the answer is negative for many uncountable cardinalities; for example, for 2^ℵ ₀, and assuming G.C.H. for every cardinal whose confinality is >ℵ₀. However, if λ>κ when κ is strongly compact and of λ=ℵ₀, then there exists a universal locally finite group of cardinality λ. The idea is to use the failure of the amalgamation property in a strong sense. We shall also prove the failure of the amalgamation property for universal locally finite groups by transferring the kind of failure of the amalgamation property from LF into ULF. We would like to thank Simon Thomas for reading carefully a preliminary version of this paper, proving Lemma 20 and making valuable remarks. Also we thank the United States—Israel Binational Science Foundation for partially supporting this work.     

A substitute for Hall's theorem for families with infinite sets

A sufficient condition for the existence of a system of distinct representatives for a family S is that x?A?S implies the number of elements of A is not smaller than the number of sets in S to which x belongs.