arXiv subjects
Publications and source records attributed to Assaf Rinot.
We address the question of whether a reflecting stationary set may be partitioned into two or more reflecting stationary subsets, providing various affirmative answers in ZFC. As an application to singular cardinals combinatorics, we infer that it is never the case that there exists a singular cardinal all of whose scales are very good.
We present a weak sufficient condition for the existence of Souslin trees at successor of regular cardinals. The result is optimal and simultaneously improves an old theorem of Gregory and a more recent theorem of the author.
The productivity of the $κ$-chain condition, where $κ$ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970s, consistent examples of $κ$-cc posets whose squares are not $κ$-cc were constructed by Laver, Galvin, Roitman and Fleissner. Later, $\mathsf{ZFC}$ examples were constructed by Todorcevic, Shelah, and others. The most difficult case, that in which $κ= \aleph_2$, was resolved by Shelah in 1997. In this work, we obtain analogous results regarding the infinite productivity of strong chain conditions, such as the Knaster property. Among other results, for any successor cardinal $κ$, we produce a $\mathsf{ZFC}$ example of a poset with precaliber $κ$ whose $ω^{\mathrm{th}}$ power is not $κ$-cc. To do so, we carry out a systematic study of colorings satisfying a strong unboundedness condition. We prove a number of results indicating circumstances under which such colorings exist, in particular focusing on cases in which these colorings are moreover closed.
We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $λ$, if $λ^{++}$ is not a Mahlo cardinal in Gödel's constructible universe, then $2^λ= λ^+$ entails the existence of a $λ^+$-complete $λ^{++}$-Souslin tree.
Ben-David and Shelah proved that if $λ$ is a singular strong-limit cardinal and $2^λ=λ^+$, then $\square^*_λ$ entails the existence of a normal $λ$-distributive $λ^+$-Aronszajn tree. Here, it is proved that the same conclusion remains valid after replacing the hypothesis $\square^*_λ$ by $\square(λ^+,{<}λ)$. As $\square(λ^+,{<}λ)$ does not impose a bound on the order-type of the witnessing clubs, our construction is necessarily different from that of Ben-David and Shelah, and instead uses walks on ordinals augmented with club guessing. A major component of this work is the study of postprocessing functions and their effect on square sequences. A byproduct of this study is the finding that for $κ$ regular uncountable, $\square(κ)$ entails the existence of a partition of $κ$ into $κ$ many fat sets. When contrasted with a classic model of Magidor, this shows that it is equiconsistent with the existence of a weakly compact cardinal that $ω_2$ cannot be split into two fat sets.
It is well-known that the square principle $\square_λ$ entails the existence of a non-reflecting stationary subset of $λ^+$, whereas the weak square principle $\square^*_λ$ does not. Here we show that if $μ^{\mathrm{cf}(λ)} < λ$ for all $μ< λ$, then $\square^*_λ$ entails the existence of a non-reflecting stationary subset of $E^{λ^+}_{\mathrm{cf}(λ)}$ in the forcing extension for adding a single Cohen subset of $λ^+$. It follows that indestructible forms of simultaneous stationary reflection entail the failure of weak square. We demonstrate this by settling a question concerning the subcomplete forcing axiom (SCFA), proving that SCFA entails the failure of $\square^*_λ$ for every singular cardinal $λ$ of countable cofinality.
We prove that reflection of the coloring number of graphs is consistent with non-reflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large gaps) is compatible with each of the following compactness principles: Rado's conjecture, Fodor-type reflection, $Δ$-reflection, Stationary-sets reflection, Martin's Maximum, and a generalized Chang's conjecture. This is accomplished by showing that, under $\mathrm{GCH}$-type assumptions, instances of incompactness for the chromatic number can be derived from square-like principles that are compatible with large amounts of compactness. In addition, we prove that, in contrast to the chromatic number, the coloring number does not admit arbitrarily large incompactness gaps.
We show that various analogs of Hindman's Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1: There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that for every $X\subseteq\mathbb R$ with $|X|=|\mathbb R|$, and every colour $γ\in\mathbb Q$, there are two distinct elements $x_0,x_1$ of $X$ for which $c(x_0+x_1)=γ$. This forms a simultaneous generalization of a theorem of Hindman, Leader and Strauss and a theorem of Galvin and Shelah. Theorem 2: For every Abelian group $G$, there exists a colouring $c:G\rightarrow\mathbb Q$ such that for every uncountable $X\subseteq G$, and every colour $γ$, for some large enough integer $n$, there are pairwise distinct elements $x_0,\ldots,x_n$ of $X$ such that $c(x_0+\cdots+x_n)=γ$. In addition, it is consistent that the preceding statement remains valid even after enlarging the set of colours from $\mathbb Q$ to $\mathbb R$. Theorem 3: Let $\circledast_κ$ assert that for every Abelian group $G$ of cardinality $κ$, there exists a colouring $c:G\rightarrow G$ such that for every positive integer $n$, every $X_0,\ldots,X_n \in[G]^κ$, and every $γ\in G$, there are $x_0\in X_0,\ldots, x_n\in X_n$ such that $c(x_0+\cdots+x_n)=γ$. Then $\circledast_κ$ holds for unboundedly many uncountable cardinals $κ$, and it is consistent that $\circledast_κ$ holds for all regular uncountable cardinals $κ$.
An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion --- Cohen forcing --- adds an $\aleph_1$-Souslin tree. In this paper, we identify a rather large class of notions of forcing that, assuming a GCH-type assumption, add a $λ^+$-Souslin tree. This class includes Prikry, Magidor and Radin forcing.
We propose a parameterized proxy principle from which $κ$-Souslin trees with various additional features can be constructed, regardless of the identity of $κ$. We then introduce the microscopic approach, which is a simple method for deriving trees from instances of the proxy principle. As a demonstration, we give a construction of a coherent $κ$-Souslin tree that applies also for $κ$ inaccessible. We then carry out a systematic study of the consistency of instances of the proxy principle, distinguished by the vector of parameters serving as its input. Among other things, it will be shown that all known $\diamondsuit$-based constructions of $κ$-Souslin trees may be redirected through this new proxy principle.
We study the relationship between a $κ$-Souslin tree $T$ and its reduced powers $T^θ/\mathcal U$. Previous works addressed this problem from the viewpoint of a single power $θ$, whereas here, tools are developed for controlling different powers simultaneously. As a sample corollary, we obtain the consistency of an $\aleph_6$-Souslin tree $T$ and a sequence of uniform ultrafilters $\langle \mathcal U_n\mid n<6\rangle$ such that $ T^{\aleph_n}/\mathcal U_n$ is $\aleph_6$-Aronszajn iff $n<6$ is not a prime number.
It is proved that in Godel's constructible universe, for every infinite successor cardinal k, there exist graphs G and H of size and chromatic number k, for which the tensor product graph (G x H) is countably chromatic.
Assume GCH and let $λ$ denote an uncountable cardinal. We prove that if $\square_λ$ holds, then this may be witnessed by a coherent sequence $< C_α| α< λ^+ >$ with the following remarkable guessing property: For every sequence $< A_i | i<λ>$ of unbounded subsets of $λ^+$, and every limit $θ<λ$, there exists some $α<λ^+$ such that $\otp(C_α)=θ$, and the $(i+1)_{th}$-element of $C_α$ is a member of $A_i$, for all $i<θ$. As an application, we construct an homogenous $λ^+$-Souslin tree from $GCH+\square_λ$, for every singular cardinal $λ$. In addition, as a by-product, a theorem of Farah and Velickovic, and a theorem of Abraham, Shelah and Solovay are generalized to cover the case of successors of regulars.
It is proved that every singular cardinal $λ$ admits a function $RTS:[λ^+]^2\rightarrow[λ^+]^2$ that transforms rectangles into squares. Namely, for every cofinal subsets $A,B$ of $λ^+$, there exists a cofinal subset $C$ of $lambda^+$, such that $RTS[AxB]$ covers CxC. When combined with a recent result of Eisworth, this shows that Shelah's notion of strong coloring $Pr_1(λ^+,λ^+,λ^+,\cf(λ))$ coincides with the classical negative partition relation $λ^+\not\rightarrow[λ^+]^2_{λ^+}$.
We survey some recent results on the validity of Jensen's diamond principle at successor cardinals. We also discuss weakening of this principle such as club guessing, and anti-diamond principles such as uniformization. A collection of open problems is included.
In their paper from 1981, Milner and Sauer conjectured that for any poset P, if cf(P)=lambda>cf(lambda)=kappa, then P must contain an antichain of size kappa. We prove that for lambda>cf(lambda)=kappa, if there exists a cardinal mu<lambda such that cov(lambda,mu,kappa,2)=lambda, then any poset of cofinality lambda contains lambda^kappa antichains of size kappa. The hypothesis of our theorem is very weak and is a consequence of many well-known axioms such as GCH, SSH and PFA. The consistency of the negation of this hypothesis is unknown.