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Monotonically far b-scale unbounded function Prevalent singular cardinals Sierpinski's onto mapping principle Small forcing transformations sap xbox Strongly Luzin set Forcing with side conditions Square-Brackets Partition Relations super-Souslin tree Ascent Path Interval topology on trees Ramsey theory over partitions Almost Souslin Cardinal Invariants ZFC construction stationary reflection Aronszajn tree Open Access Sakurai's Bell inequality S-Space Reflecting stationary set Singular cofinality reflection principles Whitehead Problem Minimal Walks Forcing Antichain Forcing Axioms Precaliber PFA stationary hitting very good scale weak Kurepa tree Chang's conjecture Diamond for trees higher Baire space Fat stationary set countably metacompact approachability ideal Well-behaved magma OCA Singular cardinals combinatorics Subtle tree property middle diamond Martin's Axiom O-space Partition Relations Shelah's Strong Hypothesis Almost countably chromatic Distributive tree Mandelbrot set Greatly Mahlo Lipschitz reduction Was Ulam right? Local Club Condensation. Luzin set ccc Souslin Tree Ascending path Generalized descriptive set theory 54G20 Fodor-type reflection tensor product graph nonmeager set Rado's conjecture Constructible Universe Poset polarized partition relation Kurepa Hypothesis Non-saturation weak square Reduced Power Knaster Hindman's Theorem Diamond-sharp Uniformly homogeneous coloring number Dushnik-Miller Generalized Clubs full tree Singular Density Chromatic number Respecting tree Club Guessing strongly bounded groups P-Ideal Dichotomy Erdos-Hajnal graphs weak diamond Axiom R Ostaszewski square Ineffable cardinal perfectly normal Subtle cardinal square Strong coloring SNR Fast club Knaster and friends Postprocessing function Successor of Regular Cardinal free Souslin tree square principles Foundations Analytic sets AIM forcing Sigma-Prikry Rock n' Roll Strongly compact cardinal Hereditarily Lindelöf space L-space GMA Commutative cancellative semigroups diamond star Large Cardinals club_AD Ulam matrix Erdos Cardinal regressive Souslin tree C-sequence Rainbow sets Iterated forcing Universal Sequences Commutative projection system Subnormal ideal Cardinal function Absoluteness Dowker space HOD Microscopic Approach stick specializable Souslin tree PFA(S)[S] indecomposable filter Weakly compact cardinal Uniformization Diamond Hedetniemi's conjecture Nonspecial tree Uniformly coherent Coherent tree Entangled linear order Almost-disjoint family projective Boolean algebra Vanishing levels Partition relations for trees Prikry-type forcing Countryman line positive partition relation Parameterized proxy principle free Boolean algebra Selective Ultrafilter Subadditive Slim tree Filter reflection Closed coloring incompactness Intersection model Successor of Singular Cardinal Cohen real Jonsson cardinal Amenable C-sequence
Author Archives: Assaf Rinot
Complicated colorings
Abstract. If $\lambda,\kappa$ are regular cardinals, $\lambda>\kappa^+$, and $E^\lambda_{\ge\kappa}$ admits a nonreflecting stationary set, then $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ holds. (Recall that $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ asserts the existence of a coloring $d:[\lambda]^2\rightarrow\lambda$ such that for any family $\mathcal A\subseteq[\lambda]^{<\kappa}$ of size $\lambda$, consisting of pairwise … Continue reading
Posted in Partition Relations, Publications
Tagged Minimal Walks, Open Access, Square-Brackets Partition Relations
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MFO workshop in Set Theory, January 2014
I gave an invited talk at the Set Theory workshop in Obwerwolfach, January 2014. Talk Title: Complicated Colorings. Abstract: If $\lambda,\kappa$ are regular cardinals, $\lambda>\kappa^+$, and $E^{\lambda}_{\ge\kappa}$ admits a nonreflecting stationary set, then $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ holds. Downloads:
Walk on countable ordinals: the characteristics
In this post, we shall present a few aspects of the method of walk on ordinals (focusing on countable ordinals), record its characteristics, and verify some of their properties. All definitions and results in this post are due to Todorcevic. … Continue reading
Polychromatic colorings
These are lectures notes of two talks Dani Livne gave in our Infinite Combinatorics seminar. I did not take notes in real-time, hence, all possible mistakes here are due to myself. Recall that a function $f:A\rightarrow B$ is said to … Continue reading
Universal binary sequences
Notation. Write $\mathcal Q(A):=\{ a\subseteq A\mid a\text{ is finite}, a\neq\emptyset\}$. Suppose for the moment that we are given a fixed sequence $\langle f_\alpha:\omega\rightarrow2\mid \alpha\in a\rangle$, indexed by some set $a$ of ordinals. Then, for every function $h:a\rightarrow\omega$ and $i<\omega$, we … Continue reading
Mathematics Colloquium, Bar-Ilan University, November 2013
I gave a colloquium talk at Bar-Ilan University on November 10, 2013. Title: Forcing as a tool to prove theorems Abstract: Paul Cohen celebrated solution to Hilbert’s first problem showed that the Continuum Hypothesis is independent of the usual axioms of … Continue reading
Syndetic colorings with applications to S and L
Notation. Write $\mathcal Q(A):=\{ a\subseteq A\mid a\text{ is finite}, a\neq\emptyset\}$. Definition. An L-space is a regular hereditarily Lindelöf topological space which is not hereditarily separable. Definition. We say that a coloring $c:[\omega_1]^2\rightarrow\omega$ is L-syndetic if the following holds. For every uncountable … Continue reading
Open coloring and the cardinal invariant $\mathfrak b$
Nik Weaver asked for a direct proof of the fact that Todorcevic’s axiom implies the failure of CH fails. Here goes. Notation. For a set $X$, we write $[X]^2$ for the set of unordered pairs $\{ \{x,x’\}\mid x,x’\in X, x\neq … Continue reading
Gabriel Belachsan (14/5/1976 – 20/8/2013)
רק כשעיני סגורות, עולם נגלה לפני
Hedetniemi’s conjecture for uncountable graphs
Abstract. It is proved that in Godel’s constructible universe, for every successor cardinal $\kappa$, there exist graphs $\mathcal G$ and $\mathcal H$ of size and chromatic number $\kappa$, for which the tensor product graph $\mathcal G\times\mathcal H$ is countably chromatic. … Continue reading
