The first omitting cardinal for Magidority

An infinite cardinal λ is Magidor if and only if . It is known that if λ is Magidor then for some , and the first such α is denoted by . In this paper we try to understand some of the properties of . We prove that can be the successor of a supercompact cardinal, when λ is a Magidor cardinal. From this result we obtain the consistency of being a successor of a singular cardinal with uncountable cofinality.     

Factorials and the finite sequences of sets

We write for the cardinality of the set of finite sequences of a set which is of cardinality . With the Axiom of Choice (), for every infinite cardinal where is the cardinality of the permutations on a set which is of cardinality . In this paper, we show that “ for every cardinal ”  is provable in and this is the best possible result in the absence of . Similar results are also obtained for : the cardinality of the set of finite sequences without repetition of a set which is of cardinality .     

Reductions on equivalence relations generated by universal sets

Let X, Y be Polish spaces, , . We say A is universal for Γ provided that each x‐section of A is in Γ and each element of Γ occurs as an x‐section of A. An equivalence relation generated by a set is denoted by , where . The following results are shown:

• (1) If A is a set universal for all nonempty closed subsets of Y, then is a equivalence relation and .
• (2) If A is a set universal for all countable subsets of Y, then is a equivalence relation, and
  • (i) and ;
  • (ii) if , then ;
  • (iii) if every set is Lebesgue measurable or has the Baire property, then .
  • (iv) for , if every set has the Baire property, and E is any equivalence relation, then .

     

On the relative strengths of fragments of collection

Let be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, Δ₀‐separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set‐theoretic collection scheme to . We focus on two common parameterisations of the collection: ‐collection, which is the usual collection scheme restricted to ‐formulae, and strong ‐collection, which is equivalent to ‐collection plus ‐separation. The main result of this paper shows that for all ,

1. proves that there exists a transitive model of Zermelo Set Theory plus ‐collection,
2. the theory is ‐conservative over the theory .

It is also shown that (2) holds for when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke‐Platek Set Theory with Infinity plus ) that does not include the powerset axiom.     

Jump inversions of algebraic structures and Σ‐definability

It is proved that for every countable structure and a computable successor ordinal α there is a countable structure which is ‐least among all countable structures such that is Σ‐definable in the αth jump . We also show that this result does not hold for the limit ordinal . Moreover, we prove that there is no countable structure with the degree spectrum for .     

A variant of Shelah's characterization of Strong Chang's Conjecture

Shelah considered a certain version of Strong Chang's Conjecture which we denote , and proved that it is equivalent to several statements, including the assertion that Namba forcing is semiproper. We introduce an apparently weaker version, denoted , and prove an analogous characterization of it. In particular, is equivalent to the assertion that the the Friedman‐Krueger poset is semiproper. This strengthens and sharpens results by Cox and sheds some light on problems posed by Usuba, Torres‐Perez and Wu.     

Some properties of infinite factorials

With the Axiom of Choice , for any infinite cardinal but, without , we cannot conclude any relationship between and for an arbitrary infinite cardinal . In this paper, we give some properties of in the absence of and compare them to those of for an infinite cardinal . Among our results, we show that “ for any infinite cardinal and any natural number n” is provable in although “ for any infinite cardinal ” is not.     

The tree property and the continuum function below

We say that a regular cardinal κ, , has the tree property if there are no κ‐Aronszajn trees; we say that κ has the weak tree property if there are no special κ‐Aronszajn trees. Starting with infinitely many weakly compact cardinals, we show that the tree property at every even cardinal , , is consistent with an arbitrary continuum function below which satisfies , . Next, starting with infinitely many Mahlo cardinals, we show that the weak tree property at every cardinal , , is consistent with an arbitrary continuum function below which satisfies , . Thus the tree property has no provable effect on the continuum function below except for the trivial requirement that the tree property at implies for every infinite κ.     

Neutrally expandable models of arithmetic

A subset of a model of is called neutral if it does not change the relation. A model with undefinable neutral classes is called neutrally expandable. We study the existence and non‐existence of neutral sets in various models of . We show that cofinal extensions of prime models are neutrally expandable, and ω₁‐like neutrally expandable models exist, while no recursively saturated model is neutrally expandable. We also show that neutrality is not a first‐order property. In the last section, we study a local version of neutral expandability.     

Indivisible sets and well‐founded orientations of the Rado graph

Every set can been thought of as a directed graph whose edge relation is ∈ . We show that many natural examples of directed graphs of this kind are indivisible: for every infinite κ, for every indecomposable λ, and every countable model of set theory. All of the countable digraphs we consider are orientations of the countable random graph. In this way we find indivisible well‐founded orientations of the random graph that are distinct up to isomorphism, and ?₁ that are distinct up to siblinghood.     

On the finite axiomatizability of

The question of whether the bounded arithmetic theories and are equal is closely connected to the complexity question of whether is equal to . In this paper, we examine the still open question of whether the prenex version of , , is equal to . We give new dependent choice‐based axiomatizations of the ‐consequences of and . Our dependent choice axiomatizations give new normal forms for the ‐consequences of and . We use these axiomatizations to give an alternative proof of the finite axiomatizability of and to show new results such as is finitely axiomatized and that there is a finitely axiomatized theory, , containing and contained in . On the other hand, we show that our theory for splits into a natural infinite hierarchy of theories. We give a diagonalization result that stems from our attempts to separate the hierarchy for .     

On cardinal characteristics of Yorioka ideals

Yorioka introduced a class of ideals (parametrized by reals) on the Cantor space to prove that the relation between the size of the continuum and the cofinality of the strong measure zero ideal on the real line cannot be decided in . We construct a matrix iteration of c.c.c. posets to force that, for many ideals in that class, their associated cardinal invariants (i.e., additivity, covering, uniformity and cofinality) are pairwise different. In addition, we show that, consistently, the additivity and cofinality of Yorioka ideals does not coincide with the additivity and cofinality (respectively) of the ideal of Lebesgue measure zero subsets of the real line.     

Generic at

In this paper we introduce a generic large cardinal akin to , together with the consequences of being such a generic large cardinal. In this case is Jónsson, and in a choiceless inner model many properties hold that are in contrast with pcf theory in .     

Algebraic numbers with elements of small height

In this paper, we study the field of algebraic numbers with a set of elements of small height treated as a predicate. We prove that such structures are not simple and have the independence property. A real algebraic integer is called a Salem number if α and are Galois conjugate and all other Galois conjugates of α lie on the unit circle. It is not known whether 1 is a limit point of Salem numbers. We relate the simplicity of a certain pair with Lehmer's conjecture and obtain a model‐theoretic characterization of Lehmer's conjecture for Salem numbers.     

Free sets for set‐mappings relative to a family of sets

Given a family of subsets of , we try to compute the least natural number n such that for every function there exists a bijection such that for all .     

On the honeymoon Oberwolfach problem

The honeymoon Oberwolfach problem HOP asks the following question. Given newlywed couples at a conference and round tables of sizes , is it possible to arrange the participants at these tables for meals so that each participant sits next to their spouse at every meal and sits next to every other participant exactly once? A solution to HOP is a decomposition of , the complete graph with additional copies of a fixed 1‐factor , into 2‐factors, each consisting of disjoint ‐alternating cycles of lengths . It is also equivalent to a semi‐uniform 1‐factorization of of type ; that is, a 1‐factorization such that for all , the 2‐factor consists of disjoint cycles of lengths . In this paper, we first introduce the honeymoon Oberwolfach problem and then present several results. Most notably, we completely solve the case with uniform cycle lengths, that is, HOP. In addition, we show that HOP has a solution in each of the following cases: ; is odd and ; as well as for all . We also show that HOP has a solution whenever is odd and the Oberwolfach problem with tables of sizes has a solution.     

On Turán problems for Cartesian products of graphs

Let be disjoint sets of sizes and . Let be a family of quadruples, having elements from and from , such that any subset with and contains one of the quadruples. We prove that the smallest size of is as . We also solve asymptotically a more general two‐partite Turán problem for quadruples.     

Dimension of the space generated by Steiner systems

To every Steiner system with parameters on and blocks , we can assign its characteristic vector , which is a ‐vector whose entries are indexed by the ‐subsets of such that for each ‐subset of if and only if . In this paper, we show that the dimension of the vector space generated by all of the characteristic vectors of Steiner systems with parameters is , provided that and there is at least one such system.     

The maximum number of columns in supersaturated designs with

Cheng and Tang [Biometrika, 88 (2001), pp. 1169–1174] derived an upper bound on the maximum number of columns that can be accommodated in a two‐symbol supersaturated design (SSD) for a given number of rows () and a maximum in absolute value correlation between any two columns (). In particular, they proved that for (mod ) and . However, the only known SSD satisfying this upper bound is when . By utilizing a computer search, we prove that for , and . These results are obtained by proving the nonexistence of certain resolvable incomplete blocks designs. The combinatorial properties of the RIBDs are used to reduce the search space. Our results improve the lower bound for SSDs with rows and columns, for , and . Finally, we show that a skew‐type Hadamard matrix of order can be used to construct an SSD with rows and columns that proves . Hence, we establish for and for all (mod ) such that . Our result also implies that when is a prime power and (mod ). We conjecture that for all and (mod ), where is the maximum number of equiangular lines in with pairwise angle .     

A note on the deductive strength of the Nielsen‐Schreier theorem

We show that the Boolean Prime Ideal Theorem () does not imply the Nielsen‐Schreier Theorem () in , thus strengthening the result of Kleppmann from “Nielsen‐Schreier and the Axiom of Choice” that the (strictly weaker than ) Ordering Principle () does not imply in . We also show that is false in Mostowski's Linearly Ordered Model of . The above two results also settle the corresponding open problems from Howard and Rubin's “Consequences of the Axiom of Choice”.