We study the structured rainbow Ramsey theory at uncountable cardinals. When compared to the usual rainbow Ramsey theory, the variation focuses on finding a rainbow subset that not only is of a certain cardinality but also satisfies certain structural constraints, such as being stationary or closed in its supremum. In the process of dealing with cardinals greater than $ω_1$, we uncover some connections between versions of Chang's Conjectures and instances of rainbow Ramsey partition relations, addressing a question raised in \cite{zhang}.
We give a negative answer to a question of Erdos and Hajnal: it is consistent that GCH holds and there is a colouring $c:[{ω_2}]^2\to 2$ establishing $ω_2 \not\to [(ω_1;ω)]^2_2$ such that some colouring $g:[ω_1]^2\to 2$ can not be embedded into $c$. It is also consistent that $2^{ω_1}$ is arbitrarily large, and a function $g$ establishes $2^{ω_1} \not\to [(ω_1,ω_2)]^2_{ω_1}$ such that there is no uncountable $g$-rainbow subset of $2^{ω_1}$. We also show that for each $k\in ω$ it is consistent with Martin's Axiom that the negative partition relation $ω_1 \not\to^* [(ω_1;ω_1)]_{k-bdd}$ holds.
The rainbow Ramsey theorem states that every coloring of tuples where each color is used a bounded number of times has an infinite subdomain on which no color appears twice. The restriction of the statement to colorings over pairs (RRT22) admits several characterizations: it is equivalent to finding an infinite subset of a 2-random, to diagonalizing against Turing machines with the halting set as oracle... In this paper we study principles that are closely related to the rainbow Ramsey theorem, the Erdős Moser theorem and the thin set theorem within the framework of reverse mathematics. We pro…
It is shown that the existence of a measurable cardinal is equiconsistent to a model of ZFC in which there is no ordinal-definable, stationary, costationary subset of $ω_1$
Here we deal with some problems posed by Matet. The first section deals with the existence of stationary subsets of [lambda]^{<kappa} with no unbounded subsets which are not stationary, where, of course, kappa is regular uncountable less or equal than lambda. In the second section we deal with the existence of such clubs. The proofs are easy but the result seems to be very surprising. Theorem 1.2 was proved some time ago by Baumgartner and is presented here for the sake of completeness.
A stationary subset S of a regular uncountable cardinal kappa reflects fully at regular cardinals if for every stationary set T subseteq kappa of higher order consisting of regular cardinals there exists an alpha in T such that S cap alpha is a stationary subset of alpha. We prove that the Axiom of Full Reflection which states that every stationary set reflects fully at regular cardinals, together with the existence of n-Mahlo cardinals is equiconsistent with the existence of Pi^1_n-indescribable cardinals. We also state the appropriate generalization for greatly Mahlo cardinals.
We prove that there is a certain degree of independence between stationary reflection phenomena at different cofinalities; e.g. it is consistent that every stationary subset of S_1^3 reflects at a point of cofinality aleph_2 while every stationary subset of S^3_0 has a non-reflecting stationary subset
If $S,T$ are stationary subsets of a regular uncountable cardinal $κ$, we say that $S$ reflects fully in $T$, $S<T$, if for almost all $α\in T$ (except a nonstationary set) $S \cap α$ is stationary in $α.$ This relation is known to be a well founded partial ordering. We say that a given poset $P$ is realized by the reflection ordering if there is a maximal antichain $\langle X_p ; p \in P \rangle$ of stationary subsets of $Reg(κ)$ so that $$\forall p,q \in P \; \forall S\subseteq X_p, T\subseteq X_q \text{ stationary}:(S<T \leftrightarrow p<_P q ) .$$ We prove that if $κ$ is $\Cal P _2 κ-$stro…
Bounded stationary reflection at a cardinal $λ$ is the assertion that every stationary subset of $λ$ reflects but there is a stationary subset of $λ$ that does not reflect at arbitrarily high cofinalities. We produce a variety of models in which bounded stationary reflection holds. These include models in which bounded stationary reflection holds at the successor of every singular cardinal $μ> \aleph_ω$ and models in which bounded stationary reflection holds at $μ^+$ but the approachability property fails at $μ$.
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 subcomplet…
We deal with the problem of preserving various versions of completeness in (< kappa) --support iterations of forcing notions, generalizing the case ``S --complete proper is preserved by CS iterations for a stationary co-stationary S subseteq omega_1''. We give applications to Uniformization and the Whitehead problem. In particular, for a strongly inaccessible cardinal kappa and a stationary set S subseteq kappa with fat complement we can have uniformization for < A_delta : delta in S'>, A_delta subseteq delta = sup A_delta, cf(delta)=otp(A_delta) and a stationary non-reflecting set S' subseteq…
A stationary subset $S$ of a regular uncountable cardinal $κ$ {\it reflects fully} at regular cardinals if for every stationary set $T \subseteq κ$ of higher order consisting of regular cardinals there exists an $α\in T$ such that $S \cap α$ is a stationary subset of $α$. {\it Full Reflection} states that every stationary set reflects fully at regular cardinals. We will prove that under a slightly weaker assumption than $κ$ having Mitchell order $κ^{++}$ it is consistent that Full Reflection holds at every $λ\leq κ$ and $κ$ is measurable.
Building on early work by Stevo Todorcevic, we describe a theory of stationary subtrees of trees of successor-cardinal height. We define the diagonal union of subsets of a tree, as well as normal ideals on a tree, and we characterize arbitrary subsets of a non-special tree as being either stationary or non-stationary. We then use this theory to prove the following partition relation for trees: Main Theorem: Let $κ$ be any infinite regular cardinal, let $ξ$ be any ordinal such that $2^{\left|ξ\right|} < κ$, and let $k$ be any natural number. Then \[ \text{non-$\left(2^{<κ}\right)$-special tree …
In this paper a hypergraph will be identified with the family of its edges. A hypergraph $\mathcal E$ possesses property $C(k,ρ)$ iff $|\bigcap \mathcal E'|<ρ$ for each $\mathcal E'\in {[\mathcal E]}^{k}$. A vertex set $Y\subset \bigcup\mathcal E$ is a "vertex cover" of $\mathcal E$ iff $E\cap Y\ne \emptyset$ for each $E\in \mathcal E$. A vertex cover $Y$ is "minimal" iff no proper subset of $Y$ is vertex cover. If $A$ is a set and $S$ is a set of cardinals, write $$ {[A]}^{S}=\{B\subset A: |B|\in S\}. $$ If $λ$ and $ρ$ are cardinals, $S$ is a set of cardinals, $k\in ω$, then we write $$\mathb…
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 prove that, e.g., if mu >cf(mu)= aleph_0 and mu>2^{aleph_0} and every stationary family of countable subsets of mu^+ reflect in some subset of mu^+ of cardinality aleph_1, then the SCH for mu^+ (moreover, for mu^+, any scale for mu^+ has a bad stationary set of cofinality aleph_1). This answers a question of Foreman and Todorcevic who got such conclusion from the simultaneous reflection of four stationary sets.
We prove that if lambda is a strong limit singular cardinal and kappa a regular uncountable cardinal < lambda, then NS_{kappa lambda}, the non-stationary ideal over P_{kappa} lambda, is nowhere precipitous. We also show that under the same hypothesis every stationary subset of P_{kappa} lambda can be partitioned into lambda^{< kappa} disjoint stationary sets.
It is consistent that for every n >= 2, every stationary subset of omega_n consisting of ordinals of cofinality omega_k where k = 0 or k <= n-3 reflects fully in the set of ordinals of cofinality omega_{n-1}. We also show that this result is best possible.
The concepts of closed unbounded (club) and stationary sets are generalised to $γ$-club and $γ$-stationary sets, which are closely related to stationary reflection. We use these notions to define generalisations of Jensen's combinatorial principles {$\Box$} and $\diamondsuit$. We define $Π^1_γ$-indescribability and use the new $\Box^γ$-sequences to extend the result of Jensen that in the constructible universe a regular cardinal is stationary reflecting if and only if it is $Π^1_1$-indescribable: we show that in $L$ a cardinal is $Π^{1}_γ $-indescribable iff it reflects $γ$-stationary sets. Mo…
We study $Σ_1(ω_1)$-definable sets (i.e. sets that are equal to the collection of all sets satisfying a certain $Σ_1$-formula with parameter $ω_1$) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is $Σ_1(ω_1)$-definable, the set of all stationary subsets of $ω_1$ is not $Σ_1(ω_1)$-definable and the complement of every $Σ_1(ω_1)$-definable Bernstein subset of ${}^{ω_1}ω_1$ is not $Σ_1(ω_1)$-definable. In contrast, we show that the existence of a Woodin cardinal is compatible …
Our original aim was, in Abelian group theory to prove the consistency of: lambda is strong limit singular and for some properties of abelian groups which are relatives of being free, the compactness in singular fails. In fact this should work for R-modules, etc. As in earlier cases part of the work is analyzing how to move between the set theory and the algebra. Set theoretically we try to force a universe which satisfies G.C.H. and diamond holds for many stationary sets but, for every regular uncountable lambda, in some sense anything which "may" hold for some stationary set, does hold for s…
