Let "ex" be the cardinality of the smallest independent family of subsets of omega (independent means that all nontrivial Boolean combinations are infinite) which cannot be extended to a homogeneous independent family. "Homogeneous" means that every finite partial map (points to points, sets to sets) that is compatible with the family is induced by a permutation of omega.
Despite its apparently ``complex'' (second order) definition, this cardinal is equal to a well-known ``simple'' cardinal invariant from Cichon's diagram.
Following D. Sobota we call a family $\mathcal F$ of infinite subsets of $\mathbb N$ a Rosenthal family if it can replace the family of all infinite subsets of $\mathbb N$ in classical Rosenthal's Lemma concerning sequences of measures on pairwise disjoint sets. We resolve two problems on Rosenthal families: every ultrafilter is a Rosenthal family and the minimal size of a Rosenthal family is exactly equal to the reaping cardinal $\mathfrak r$. This is achieved through analyzing nowhere reaping families of subsets of $\mathbb N$ and through applying a paving lemma which is a consequence of a p…
We investigate which filters on $ω$ can contain towers, that is, a modulo finite descending sequence without any pseudointersection (in $[ω]^ω$). We prove the following results: - Many classical examples of nice tall filters contain no towers (in ZFC). - It is consistent that tall analytic P-filters contain towers of arbitrary regular height (simultaneously for many regular cardinals as well). - It is consistent that all towers generate non-meager filters, in particular (consistently) Borel filters do not contain towers. - The statement "Every ultrafilter contains towers." is independent of ZF…
This is the Ph.D. thesis of the author, which was written under the supervision of Michael Hrušák at UNAM. The main contributions of this thesis are the following: There is a $+$-Ramsey \textsf{MAD} family. This answers an old question of Michael Hrušák. There are no $P$-points in the Silver model, answering a question of Michael Hrušák (this is joint work with David Chodounský. The statement \textquotedblleft There are no $P$-points\textquotedblright\ is consistent with the continuum being arbitrarily large, this answers an open question regarding $P$-points. Every Miller indestructible \text…
A homogeneous family of subsets over a given set is one with a very ``rich'' automorphism group. We prove the existence of a bi-universal element in the class of homogeneous families over a given infinite set and give an explicit construction of $2^\cont$ isomorphism types of homogeneous families over a countable set.
For x and y sequences of real numbers define the inner product (x,y) = x(0)y(0) + x(1)y(1)+ ... which may not be finite or even exist. We say that x and y are orthogonal iff (x,y) converges and equals 0. Define l_p to be the set of all real sequences x such that |x(0)|^p + |x(1)|^p + .. converges. For Hilbert space, l_2, any family of pairwise orthogonal sequences must be countable. Thm 1. There exists a pairwise orthogonal family F of size continuum such that F is a subset of l_p for every p>2. It was already known that there exists a family of continuum many pairwise orthogonal elements of r…
We introduce the property ``$F$-linked'' of subsets of posets for a given free filter $F$ on the natural numbers, and define the properties ``$μ$-$F$-linked'' and ``$θ$-$F$-Knaster'' for posets in a natural way. We show that $θ$-$F$-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families. Concerning iterations of such posets, we develop a general technique to construct $θ$-$\mathrm{Fr}$-Knaster posets (where $\mathrm{Fr}$ is the Frechet ideal) via matrix iterations of ${<}θ$-ultrafilter-linked posets (restricted to some level of the matrix). This is a…
Our results in this paper increase the model-theoretic precision of a widely used method for building ultrafilters, and so advance the general problem of constructing ultrafilters whose ultrapowers have a precise degree of saturation. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, {thus} saturating any stable theory. We then prove directly that a "bottleneck" in the inductive construction of a regular ultrafilter on $λ$ (i.e. a point after which all antichains of $P(λ)/D$ have cardinality less than $λ$) essentiall…
The relations M(kappa,lambda,mu)->B [resp. B(sigma)] meaning that if A subset [kappa]^lambda with |A|=kappa is mu-almost disjoint then A has property B [resp. has a sigma-transversal] had been introduced and studied under GCH by Erdos and Hajnal in 1961. Our two main results here say the following: Assume GCH and rho be any regular cardinal with a supercompact [resp. 2-huge] cardinal above rho. Then there is a rho-closed forcing P such that, in V^P, we have both GCH and M(rho^{(+rho+1)},rho^+,rho) not-> B [resp. M(rho^{(+rho+1)},lambda,rho) not-> B(rho^+) for all lambda =< rho^{(+rho+1)}].
We investigate the existence of perfect homogeneous sets for analytic colorings. An analytic coloring of X is an analytic subset of [X]^N, where N>1 is a natural number. We define an absolute rank function on trees representing analytic colorings, which gives an upper bound for possible cardinalities of homogeneous sets and which decides whether there exists a perfect homogeneous set. We construct universal sigma-compact colorings of any prescribed rank gamma<omega_1. These colorings consistently contain homogeneous sets of cardinality aleph_gamma but they do not contain perfect homogeneous se…
We describe a (3,3)-homogeneous orthomodular posets for some cardinality of their sets of atoms. We examine a state space and a set of two-valued states of such logics. Particular homogeneous OMPs with exactly k pure states (k=1,...,7, 10,11)
This paper deals with variety of problems in pcf theory and infinitary combinatorics. We look at normal filters and prc, measures of the size of [lambda]^{<kappa}, pcf-inaccessibility, entangled orders (and narrow Boolean Algebras), measuring prod_{i<kappa}f(i) by a family of ideals and a family of sequences <B_i:i<kappa> with |B_i|<mu_i, and finally we deal with the existence of strongly almost disjoint families.
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 terms belonging to U. For the functions f,g from {0,1,2,...} to {0,1,2,...} we define: f =< g if and only if {i: f(i) > g(i)} is finite. Let b denote the smallest cardinality of a unbounded (in the sense of =<) family of functions from {0,1,2,...} to {0,1,2,...}, see [1]. Theorem 1. If F is a family of sequences converging to +infinity and card F < b, then …
We show that it is consistent that the continuum is as large as you wish, and for each uncountable cardinal $κ$ below the continuum, there are a subset $T$ of the reals and a family $A$ of countable subsets of $T$ such that (1) both $T$ and $A$ have cardinality $κ$, (2) $|\bar{a}\cap T|=κ$ for each $a\in A$, (3) for each uncountable subset of $T$ contains some elements of $A$, and so (i) there is an almost disjoint family of subsets of the reals with size and chromatic number $κ$, (ii) there is a locally compact, locally countable $T_2$ space with cardinality spectrum $\{ω,κ\}$.
We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness (a locality property for orbital types), and are stable (in terms of the number of orbital types) in some cardinal. Assuming the singular cardinal hypothesis (SCH), we prove a full characterization of the (high-enough) stability cardinals, and connect the stability spectrum with the behavior of saturated models. We deduce (in ZFC) that if a class is stable on a tail of cardinals, then it has no long splitting chains (the converse is known). This indicates that there is a clear not…
We give a combinatorial characterization of when a maximal almost disjoint family of a weakly compact cardinal $κ$ is indestructible by the higher random forcing $\mathbb Q_κ$. We then use this characterisation to show that $\mathrm{add}(\mathbf{null}_κ) = \mathfrak b_κ= \mathfrak c_κ$ implies the existence $\mathbb Q_κ$-indestructible family. The results and proofs presented here are parallel to those for classical random forcing.
We study maximal independent families (m.i.f.) in the projective hierarchy. We show that (a) the existence of a $\boldsymbolΣ^1_2$ m.i.f. is equivalent to the existence of a $\boldsymbolΠ^1_1$ m.i.f., (b) in the Cohen model, there are no projective maximal independent families, and (c) in the Sacks model, there is a $\boldsymbolΠ^1_1$ m.i.f. We also consider a new cardinal invariant related to the question of destroying or preserving maximal independent families.
Miller's 1937 splitting theorem was proved for pairs of cardinals $(\n,ρ)$ in which $n$ is finite and $ρ$ is infinite. An extension of Miller's theorem is proved here in ZFC for pairs of cardinals $(ν,ρ)$ in which $ν$ is arbitrary and $ρ\ge \beth_\om(ν)$. The proof uses a new general method that is based on Shelah's revises Generalized Continuum Hypothesis theorem. Upper bounds on conflict-free coloring numbers of families of sets and a general comparison theorem follow as corollaries of the main theorem. Other corollaries eliminate the use of additional axioms from splitting theorems due to E…
I introduced the notions of proper and piecewise proper families of reals to make progress on an open question in the field of models of PA about whether every Scott set is the standard system of a model of PA. A family of reals X is proper if it is arithmetically closed and the quotient Boolean algebra X/fin is a proper poset. A family is piecewise proper if it is the union of a chain of proper families of size $\leqω_1$. Here, I explore the question of the existence of proper and piecewise proper families of reals of different cardinalities.
We study density requirements on a given Banach space that guarantee the existence of subsymmetric basic sequences by extending Tsirelson's well-known space to larger index sets. We prove that for every cardinal $κ$ smaller than the first Mahlo cardinal there is a reflexive Banach space of density $κ$ without subsymmetric basic sequences. As for Tsirelson's space, our construction is based on the existence of a rich collection of homogeneous families on large index sets for which one can estimate the complexity on any given infinite set. This is used to describe detailedly the asymptotic struc…
Two sets are said to be almost disjoint if their intersection is finite. Almost disjoint subsets of [omega]^omega and omega^omega have been studied for quite some time. In particular, the cardinal invariants a and a_e, defined to be the minimum cardinality of a maximal infinite almost disjoint family of [omega]^omega and omega^omega respectively, are known to be consistently less than continuum. Here we examine analogs for functions in R^omega and projections on l^2, showing that they too can be consistently less than continuum.
The Noetherian type of a space is the least $κ$ such that it has a base that is $κ$-like with respect to containment. Just as all known homogeneous compacta have cellularity at most $2^ω$, they satisfy similar upper bounds in terms of Noetherian type and related cardinal functions. We prove these and many other results about these cardinal functions. For example, every homogeneous dyadic compactum has Noetherian type $ω$. Assuming GCH, every point in a homogeneous compactum $X$ has a local base that is $c(X)$-like with respect to containment. If every point in a compactum has a well-quasiorder…
Motivated by recent results and questions of D. Raghavan and S. Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We show that if $κ=λ^+$ for some $λ\geq ω$ and $\mathfrak b(κ)=κ^+$ then $\mathfrak a_e(κ)=\mathfrak a_p(κ)=κ^+$. If, additionally, $2^{<λ}=λ$ then $\mathfrak a_g(κ)=κ^+$ as well. Furthermore, we prove a variety of new bounds for $\mathfrak d(κ)$ in terms of $\mathfrak r(κ)$, including $\mathfrak d(κ)\leq \mathfrak r_σ(κ)\leq \mathrm{cof}([\mathfrak r(κ)]^ω)$, and…
We prove several reflection theorems on $D$-spaces, which are Hausdorff topological spaces $X$ in which for every open neighbourhood assignment $U$ there is a closed discrete subspace $D$ such that \[ \bigcup\{U(x): x\in D\}=X. \] The upwards reflection theorems are obtained in the presence of a forcing axiom, while most of the downwards reflection results use large cardinal assumptions. The combinatorial content of arguments showing that a given space is a $D$-space, can be formulated using the concept of discrete families. We note the connection between non-reflection arguments involving dis…
A ``k-rule" is a sequence A=((A_n,B_n):n<omega) of pairwise disjoint sets B_n, each of cardinality at most k, where A_n is a subset of B_n. A set X of natural numbers (a ``real'') follows a rule A if for infinitely many n we have that the intersection of X with B_n is exactly A_n. There are obvious cardinal invariants resulting from this definition: the least number of reals needed to follow all k-rules, s_k, and the least number of k-rules without a real following all of them, r_k. We investigate these cardinal invariants and their connection to some well-known cardinals from Cichon's diagram…
We prove several reflection theorems on $D$-spaces, which are Hausdorff topological spaces $X$ in which for every open neighbourhood assignment $U$ there is a closed discrete subspace $D$ such that \[ \bigcup\{U(x): x\in D\}=X. \] The upwards reflection theorems are obtained in the presence of a forcing axiom, while most of the downwards reflection results use large cardinal assumptions. The combinatorial content of arguments showing that a given space is a $D$-space, can be formulated using the concept of discrete families. We note the connection between non-reflection arguments involving dis…
Using Shelah's revised GCH theorem we prove that if mu<beth_omega <= lambda are cardinals, then every mu-almost disjoint subfamily B of [lambda]^{beth_omega} is essentially disjoint, i.e. for each b from B there is a subset f(b) of b of size < |b| such that the family {b-f(b) b in B} is disjoint. We also show that if mu<=kappa<=lambda, and kappa is infinite, and (x) every mu-almost disjoint subfamily of [lambda]^kappa is essentially disjoint, then (xx) every mu-almost disjoint family B of subsets of lambda with |b|>=kappa for all b from B has a conflict-free colorings with kappa colors. Puttin…
We survey results regarding the definability and size of maximal discrete sets in analytic hypergraphs. Our main examples include maximal almost disjoint (or mad) families, $\mathcal I$-mad families, maximal eventually different families, and maximal cofinitary groups. We discuss the non-increasing sequence of cardinal characteristics $\mathfrak a_ξ$, for $ξ<ω_1$ as well as the notions of spectra of characteristics and optimal projective witnesses. We give an account of Zhang's forcing to add generic cofinitary permutations, and of a version of this forcing with built-in coding.
Our main result is that, given a collection $\mathcal{R}$ of meager relations on a Polish space $X$ such that $|\mathcal{R}|\leqω$, there exists a dense Baire subspace $F$ of $X$ (equivalently, a nowhere meager subset $F$ of $X$) such that $F$ is $R$-free for every $R\in\mathcal{R}$. This generalizes a recent result of Banakh and Zdomskyy. As an application, we show that there exists a non-meager independent family on $ω$, and define the corresponding cardinal invariant. Furthermore, assuming Martin's Axiom for countable posets, our result can be strengthened by substituting "$|\mathcal{R}|\le…
The second author has recently shown ([20]) that any selectively (a) almost disjoint family must have cardinality strictly less than $2^{\alpeh_0}$, so under the Continuum Hypothesis such a family is necessarily countable. However, it is also shown in the same paper that $2^{\alpeh_0} < 2^{\alpeh_1}$ alone does not avoid the existence of uncountable selectively (a) almost disjoint families. We show in this paper that a certain effective parametrized weak diamond principle is enough to ensure countability of the almost disjoint family in this context. We also discuss the deductive strength of t…
