Related papers: Splitting families of sets in ZFC
It is shown that it is consistent with ZFC that every uncountable set can be continuously mapped onto a splitting family.
We investigate an extension of ZFC set theory (in an extended language) that stipulates the existence of a proper class of indiscernibles over the universe. One of the main results of the paper shows that the purely set-theoretical…
Let A_1,...,A_k be a collection of families of subsets of an n-element set. We say that this collection is cross-intersecting if for any i,j in [k] with i not equal to j, A in A_i and B in A_j implies that the intersection of A and B is…
In this paper, we consider certain cardinals in ZF (set theory without AC, the Axiom of Choice). In ZFC (set theory with AC), given any cardinals C and D, either C <= D or D <= C. However, in ZF this is no longer so. For a given infinite…
A $k$-wise $\ell$-divisible set family is a collection $\mathcal{F}$ of subsets of ${ \{1,\ldots,n \} }$ such that any intersection of $k$ sets in $\mathcal{F}$ has cardinality divisible by $\ell$. If $k=\ell=2$, it is well-known that…
Separating hash families are useful combinatorial structures which are generalizations of many well-studied objects in combinatorics, cryptography and coding theory. In this paper, using tools from graph theory and additive number theory,…
A set system F is intersecting if any pair of sets in F have a nonempty intersection. A fundamental theorem of Erd\H{o}s, Ko and Rado states that if F is an intersecting family of r-subsets of [n]={1,...,n}, and n>= 2r, then the cardinality…
G\"odel proved in the 1930s in his famous Incompleteness Theorems that not all statements in mathematics can be proven or disproven from the accepted ZFC axioms. A few years later he showed the celebrated result that Cantor's Continuum…
We consider families F of sequences converging to +infinity that F satisfies the following condition (C): (C): if an open set U in the real line is unbounded above then there exists a sequence belonging to F, which has an infinite number of…
A set theory is developed based on the approximations of sets and denoted by AS. In AS the set of all sets exists but the argument for Russell's and Cantor's paradox fail. The Axioms of Separation, Replacement and Foundation are not valid.…
We formulate and prove (in {\sf ZFC}) a strong coloring theorem which holds at successors of singular cardinals, and use it to answer several questions concerning Shelah's principle $Pr_1(\mu^+,\mu^+,\mu^+,\cf(\mu))$ for singular $\mu$.
The work in this article is concerned with two different types of families of finite sets: separating families and splitting families (they are also called "systems"). These families have applications in combinatorial search, coding theory,…
G\"odel's first and second incompleteness theorems are corner stones of modern mathematics. In this article we present a new proof of these theorems for ZFC and theories containing ZFC, using Chaitin's incompleteness theorem and a very…
Generalizing the notion of a tight almost disjoint family, we introduce the notions of a {\em tight eventually different} family of functions in Baire space and a {\em tight eventually different set of permutations} of $\omega$. Such sets…
Fairly deep results of Zermelo-Frenkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is K*K = K,…
Harvey Friedman, in his remarkable paper Finite functions and the necessary use of large cardinals, Ann. Math. 148:803-893, 1998 and in a technical report, Applications of large cardinals to graph theory, Ohio State University, 1997,…
Our theme is that not every interesting question in set theory is independent of $ZFC$. We give an example of a first order theory $T$ with countable $D(T)$ which cannot have a universal model at $\aleph_1$ without CH; we prove in $ZFC$ a…
We work in the setting of Zermelo-Fraenkel set theory without assuming the Axiom of Choice. We consider sets with the Boolean operations together with the additional structure of comparing cardinality (in the Cantorian sense of injections).…
Modulo the existence of large cardinals, there is a model of set theory in which for some set $B$ of regular cardinals, the sequence $\langle \text{pcf}^\alpha(B): \alpha \in \text{Ord} \rangle$ is strictly increasing. The result answers a…
In this paper we present a simple unifying approach to prove several statements about intersecting and cross-intersecting families, including the Erd\H os--Ko--Rado theorem, the Hilton--Milner theorem, a theorem due to Frankl concerning the…