arXiv subjects

Dilip Raghavan

Publications and source records attributed to Dilip Raghavan.

29 records · Page 2Linked to original sources

arXiv2015

Two inequalities between cardinal invariants

We prove two $\mathrm{ZFC}$ inequalities between cardinal invariants. The first inequality involves cardinal invariants associated with an analytic P-ideal, in particular the ideal of subsets of $ω$ of asymptotic density $0$. We obtain an upper bound on the $\ast$-covering number, sometimes also called the weak covering number, of this ideal by proving in Section \ref{sec:covz0} that ${\mathord{\mathrm{cov}}}^{\ast}({\mathcal{Z}}_{0}) \leq \mathfrak{d}$. In Section \ref{sec:skbk} we investigate the relationship between the bounding and splitting numbers at regular uncountable cardinals. We prove in sharp contrast to the case when $κ= ω$, that if $κ$ is any regular uncountable cardinal, then ${\mathfrak{s}}_κ \leq {\mathfrak{b}}_κ$.

math.LO↗

arXiv2014

On embedding certain partial orders into the P-points under RK and Tukey reducibility

The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin's axiom for $σ$-centered posets. In his 1973 paper he showed under this assumption that both $ω_{1}$ and the reals can be embedded. This result was later repeated for the coarser notion of Tukey reducibility. We prove in this paper that Martin's axiom for $σ$-centered posets implies that every partial order of size at most continuum can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility.

math.LO↗

arXiv2013

The next best thing to a P-point

We study ultrafilters on $ω^2$ produced by forcing with the quotient of $\scr P(ω^2)$ by the Fubini square of the Fréchet filter on $ω$. We show that such an ultrafilter is a weak P-point but not a P-point and that the only non-principal ultrafilters strictly below it in the Rudin-Keisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest square-bracket partition relations that are possible for a non-P-point. We show that it is not basically generated but that it shares with basically generated ultrafilters the property of not being at the top of the Tukey ordering. In fact, it is not Tukey-above $[ω_1]^{<ω}$, and it has only continuum many ultrafilters Tukey-below it. A tool in our proofs is the analysis of similar (but not the same) properties for ultrafilters obtained as the sum, over a selective ultrafilter, of non-isomorphic selective ultrafilters.

math.LO↗

arXiv2013

Combinatorial dichotomies and cardinal invariants

Assuming the P-ideal dichotomy, we attempt to isolate those cardinal characteristics of the continuum that are correlated with two well-known consequences of the proper forcing axiom. We find a cardinal invariant $\mathfrak{x}$ such that the statement that $\mathfrak{x} > ω_{1}$ is equivalent to the statement that 1, $ω$, $ω_{1}$, $ω\times ω_{1}$, and ${\left[ω_{1}\right]}^{< ω}$ are the only cofinal types of directed sets of size at most ${\aleph}_{1}$. We investigate the corresponding problem for the partition relation $ω_{1} \rightarrow (ω_{1}, α)^2$ for all $α< ω_{1}$. To this effect, we investigate partition relations for pairs of comparable elements of a coherent Suslin tree $\mathbb{S}$. We show that a positive partition relation for such pairs follows from the maximal amount of the proper forcing axiom compatible with the existence of $\mathbb{S}$. As a consequence we conclude that after forcing with the coherent Suslin tree $\mathbb{S}$ over a ground model satisfying this relativization of the proper forcing axiom, $ω_{1} ~\rightarrow {(ω_{1}, α)}^{2}$ for all $α< ω_{1}$. We prove that this positive partition relation for $\mathbb{S}$ cannot be improved by showing in $\mathrm{ZFC}$ that $\mathbb{S} \not\rightarrow ({\aleph}_{1}, ω+2)^2$.

math.LO↗

arXiv2012

Bounding, splitting, and almost disjointness

We investigate some aspects of bounding, splitting, and almost disjointness. In particular, we investigate the relationship between the bounding number, the closed almost disjointness number, splitting number, and the existence of certain kinds of splitting families.

math.LO↗

arXiv2012

The generic ultrafilter added by ${(\FIN \times \FIN)}^{+}$

We investigate the Tukey type of the generic ultrafilter added by the quotient $\mathcal{P}(ω\times ω) / (\mathrm{FIN} \times \mathrm{FIN})$. We prove that this ultrafilter is not basically generated and yet does not have the maximal Tukey type among directed partial orders of size continuum. Moreover, any Tukey reduction from this ultrafilter to any other ultrafilter is witnessed by a Baire class one map.

math.LO↗

arXiv2012

Splitting families and complete separability

We answer a question from Raghavan and Stepr{ā}ns' paper on weakly tight families by showing that $\mathfrak{s} = {\mathfrak{s}}_{ω, ω}$. Then we use this to construct a completely separable maximal almost disjoint family under $\s \leq \a$, partially answering a question of Shelah.

math.LO↗

arXiv2011

Comparing the closed almost disjointness and dominating numbers

We prove that if there is a dominating family of size ${\aleph}_{1}$, then there is are ${\aleph}_{1}$ many compact subsets of $ω^ω$ whose union is a maximal almost disjoint family of functions that is also maximal with respect to infinite partial functions.

math.LO↗

arXiv2010

On weakly tight families

Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when $\c < {\aleph}_ω$, we construct a weakly tight family under the hypothesis $\s \leq \b < {\aleph}_ω$. The case when $\s < \b$ is handled in $\ZFC$ and does not require $\b < {\aleph}_ω$, while an additional PCF type hypothesis, which holds when $\b < {\aleph}_ω$ is used to treat the case $\s = \b$. The notion of a weakly tight family is a natural weakening of the well studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by Hru{š}{á}k and Garc{\'ı}a Ferreira \cite{Hr1}, who applied it to the Katétov order on almost disjoint families.

math.LO↗

arXiv2008

Gregory Trees, The Continuum, And Martin's Axiom

We continue the investigation of Gregory trees and the Cantor Tree Property carried out by Hart and Kunen. We produce models of MA with the Continuum arbitrarily large in which there are Gregory trees, and in which there are no Gregory trees.

math.LO↗

arXiv2008

There is a Van Douwen MAD family

We prove in ZFC that there is a MAD family of functions in omega^omega which is also maximal with respect to infinite partial functions. This solves a 20 year old question of Van Douwen. We also strengthen a result of J. Steprans stating that strongly MAD families of functions cannot be analytic. We show that analytic MAD families of functions, if they exist, must satisfy some strong constraints.

math.LO↗