Q-points in the Tukey order
Q-points are cofinal in the RK-ordering under several mild hypotheses.
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arXiv subjects
Dilip Raghavan
Publications and source records attributed to Dilip Raghavan.
Borel Order Dimension
We introduce and study a notion of Borel order dimension for Borel quasi orders. It will be shown that this notion is closely related to the notion of Borel dichromatic number for simple directed graphs. We prove a dichotomy, which generalizes the ${\GGG}_{0}$-dichotomy, for the Borel dichromatic number of Borel simple directed graphs. By applying this dichotomy to Borel quasi orders, another dichotomy that characterizes the Borel quasi orders of uncountable Borel dimension is proved. We obtain further structural information about the Borel quasi orders of countable Borel dimension by showing that they are all Borel linearizable. We then investigate the locally countable Borel quasi orders in more detail, paying special attention to the Turing degrees, and produce models of set theory where the continuum is arbitrarily large and all locally countable Borel quasi orders are of Borel dimension less than the continuum. Combining our results here with earlier work shows that the Borel order dimension of the Turing degrees is usually strictly larger than its classical order dimension.
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Adding ultrafilters to Shelah's model for no nowhere dense ultrafilters
We exhibit a forcing for producing a model with no nowhere dense ultrafilters that satisfies the full Sacks Property. By interleaving this forcing with other forcing notions, a model containing a $(2, {\aleph}_{0})$-selective ultrafilter, but no nowhere dense ultrafilters is produced. It is thus proved that the existence of $(2, {\aleph}_{0})$-selective ultrafilters does not imply the existence of nowhere dense ultrafilters.
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Order structure of P-point ultrafilters and their relatives
We survey some recent results about the order structure of various kinds of ultrafilters. More precisely, we study Rudin-Keisler and Tukey reducibility in classes of selective, stable ordered-union, and P-point ultrafilters. Although these reductions are fundamentally different, there are connections between them. On the other hand, even though the classes of ultrafilters we consider are similar, there are significant differences in their order structure, as will be seen in the survey.
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Stable ordered-union versus selective ultrafilters
It will be shown to be consistent that there are at least two non-isomorphic selective ultrafilters, but no stable ordered-union ultrafilters. This answers a question of Blass from his 1987 paper which introduced the concept of a stable ordered-union ultrafilter.
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Combinatorial properties of MAD families
We study some strong combinatorial properties of $\textsf{MAD}$ families. An ideal $\mathcal{I}$ is Shelah-Stepr\={a}ns if for every set $X\subseteq{\left[ \omega\right]}^{<\omega}$ there is an element of $\mathcal{I}$ that either intersects every set in $X$ or contains infinitely many members of it. We prove that a Borel ideal is Shelah-Stepr\={a}ns if and only if it is Kat\v{e}tov above the ideal $\textsf{fin}\times\textsf{fin}$. We prove that Shelah-Stepr\={a}ns $\textsf{MAD}$ families have strong indestructibility properties (in particular, they are both Cohen and random indestructible). We also consider some other strong combinatorial properties of $\textsf{MAD}$ families. Finally, it is proved that it is consistent to have $\mathrm{non}(\mathcal{M}) = {\aleph}_{1}$ and no Shelah-Stepr\={a}ns families of size ${\aleph}_{1}$.
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Lower bounds of sets of P-points
We show that $\mathrm{MA}_{\kappa}$ implies that each collection of ${P}_{\mathfrak c}$-points of size at most $\kappa$ which has a $P_{\mathfrak c}$-point as an $RK$ upper bound also has a ${P}_{\mathfrak c}$-point as an $RK$ lower bound.
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A combinatorial property of rho-functions
We show that if $\mathcal{T}$ is any Hausdorff topology on ${\omega}_{1}$, then any subset of ${\omega}_{1}$ which is homeomorphic to the rationals under $\mathcal{T}$ can be refined to a homeomorphic copy of the rationals on which $\bar{\rho}$ is shift-increasing.
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Galvin's problem in higher dimensions
It is proved that for each natural number $n$, if $\left| \mathbb{R} \right| = {\aleph}_{n}$, then there is a coloring of ${\left[ \mathbb{R} \right]}^{n+2}$ into ${\aleph}_{0}$ colors that takes all colors on ${\left[ X \right]}^{n+2}$ whenever $X$ is any set of reals which is homeomorphic to $\mathbb{Q}$. This generalizes a theorem of Baumgartner and sheds further light on a problem of Galvin from the 1970s. Our result also complements and contrasts with our earlier result saying that any coloring of ${\left[ \mathbb{R} \right]}^{2}$ into finitely many colors can be reduced to at most $2$ colors on the pairs of some set of reals which is homeomorphic to $\mathbb{Q}$ when large cardinals exist.
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The almost disjointness invariant for products of ideals
The almost disjointness numbers associated to the quotients determined by the transfinite products of the ideal of finite sets are investigated. A $\mathrm{ZFC}$ lower bound involving the minimum of the classical almost disjointness and splitting numbers is proved for these characteristics. En route, it is shown that the splitting numbers associated to these quotients are all equal to the classical splitting number. Finally, it is proved to be consistent that the almost disjointness numbers associated to these quotients are all equal to the second uncountable cardinal while the bounding number is the first uncountable cardinal. Several open problems are considered.
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Supersaturated ideals
A $σ$-ideal $\cal{I}$ on a set $X$ is supersaturated if for every family $\cal{F}$ of $\cal{I}$-positive sets with $|\cal{F}| < \mathrm{add}(\cal{I})$, there exists a countable set that meets every set in $\cal{F}$. We show that many well-known ccc forcings preserve supersaturation. We also show that the existence of supersaturated ideals is independent of ZFC plus "There exists an $ω_1$-saturated $σ$-ideal".
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A small ultrafilter number at smaller cardinals
It is proved to be consistent relative to a measurable cardinal that there is a uniform ultrafilter on the real numbers which is generated by fewer than the maximum possible number of sets. It is also shown to be consistent relative to a supercompact cardinal that there is a uniform ultrafilter on ${\aleph}_{ω+1}$ which is generated by fewer than ${2}^{{\aleph}_{ω+1}}$ sets.
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Chains of P-points
It is proved that the Continuum Hypothesis implies that any sequence of rapid P-points of length $<{\mathfrak c}^{+}$ which is increasing with respect to the Rudin-Keisler ordering is bounded above by a rapid P-point. This is an improvement of a result from [Kuzeljević, Raghavan: A long chain of P-points, arxiv:1607.07188 [math.LO]]. It is also proved that the notion of a $δ$-generic sequence is equivalent to an apparently much weaker notion. This allows the central definition used in the construction in [Kuzeljević, Raghavan: A long chain of P-points, arxiv:1607.07188 [math.LO]] to be considerably simplified.
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Proof of a Conjecture of Galvin
We prove that if the set of unordered pairs of real numbers is colored by finitely many colors, there is a set of reals homeomorphic to the rationals whose pairs have at most two colors. Our proof uses large cardinals and it verifies a conjecture of Galvin from the 1970s. We extend this result to an essentially optimal class of topological spaces in place of the reals.
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Two results on cardinal invariants at uncountable cardinals
We prove two ZFC theorems about cardinal invariants above the continuum which are in sharp contrast to well-known facts about these same invariants at the continuum. It is shown that for an uncountable regular cardinal $κ$, $\mathfrak{b}(κ) = κ^{+}$ implies $\mathfrak{a}(κ) = κ^{+}$. This improves an earlier result of Blass, Hyttinen, and Zhang. It is also shown that if $κ\geq {\beth}_ω$ is an uncountable regular cardinal, then $\mathfrak{d}(κ) \leq \mathfrak{r}(κ)$. This result partially dualizes an earlier theorem of the authors.
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More on the density zero ideal
The main result of this paper is an improvement of the upper bound on the cardinal invariant ${\mathord{\mathrm{cov}}}^{\ast}({\mathcal{Z}}_{0})$ that was discovered by Raghavan and Shelah in an earlier paper. Here ${\mathcal{Z}}_{0}$ is the ideal of subsets of the set of natural numbers that have asymptotic density zero. This improved upper bound is also dualized to get a better lower bound on the cardinal ${\mathord{\mathrm{non}}}^{\ast}({\mathcal{Z}}_{0})$. En route some variations on the splitting number are introduced and several relationships between these variants are proved.
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A long chain of P-points
The notion of a $δ$-generic sequence of P-points is introduced in this paper. It is proved assuming the Continuum Hypothesis that for each $δ< ω_{2}$, any $δ$-generic sequence of P-points can be extended to an $ω_{2}$-generic sequence. This shows that the Continuum Hypothesis implies that there is a chain of P-points of length ${\mathfrak{c}}^{+}$ with respect to both Rudin-Keisler and Tukey reducibility. The proofs can be easily adapted to get such a chain of length ${\mathfrak{c}}^{+}$ under a more general hypothesis like Martin's Axiom. These results answer an old question of Andreas Blass.
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Suslin trees, the bounding number, and partition relations
We investigate the unbalanced ordinary partition relations of the form $λ\rightarrow {(λ, α)}^{2}$ for various values of the cardinal $λ$ and the ordinal $α$. For example, we show that for every infinite cardinal $κ,$ the existence of a $κ^{+}-$Suslin tree implies $κ^{+} \not\rightarrow {\left( κ^{+}, {\log}_κ(κ^{+}) + 2 \right)}^{2}$. The consistency of the positive partition relation $\mathfrak{b} \rightarrow {(\mathfrak{b}, α)}^{2}$ for all $α< ω_{1}$ for the bounding number $\mathfrak{b}$ is also established from large cardinals.
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