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Hedetniemi's conjecture coloring number HOD Closed coloring Strong coloring Dowker space Was Ulam right? Forcing with side conditions Square-Brackets Partition Relations Selective Ultrafilter Partition relations for trees weak Kurepa tree Dushnik-Miller Diamond ccc Prikry-type forcing Non-saturation middle diamond polarized partition relation stationary hitting Subtle cardinal Kurepa Hypothesis Ostaszewski square Lipschitz reduction Axiom R Absoluteness Minimal Walks free Boolean algebra Chromatic number Iterated forcing ZFC construction stick Almost countably chromatic Cardinal function PFA(S)[S] Rainbow sets regressive Souslin tree Fast club Open Access very good scale Reflecting stationary set full tree Nonspecial tree Ramsey theory over partitions Ascending path club_AD S-Space weak square Commutative projection system Knaster Diamond-sharp Aronszajn tree Fat stationary set Prevalent singular cardinals Ascent Path Uniformly coherent Hindman's Theorem Greatly Mahlo Club Guessing higher Baire space Ineffable cardinal Analytic sets Sierpinski's onto mapping principle Hereditarily Lindelöf space Sigma-Prikry Sakurai's Bell inequality Luzin set Filter reflection Monotonically far Chang's conjecture Poset Jonsson cardinal Uniformization C-sequence square Subtle tree property sap countably metacompact stationary reflection Uniformly homogeneous Local Club Condensation. Strongly compact cardinal GMA Countryman line tensor product graph Distributive tree Generalized Clubs Forcing Cohen real Mandelbrot set Almost-disjoint family Constructible Universe incompactness Successor of Singular Cardinal Postprocessing function P-Ideal Dichotomy projective Boolean algebra perfectly normal L-space Singular Density Well-behaved magma reflection principles Cardinal Invariants xbox super-Souslin tree Subadditive AIM forcing transformations Interval topology on trees Intersection model Entangled linear order Singular cardinals combinatorics Partition Relations nonmeager set Forcing Axioms Weakly compact cardinal Slim tree Knaster and friends Erdos Cardinal strongly bounded groups Successor of Regular Cardinal positive partition relation indecomposable filter unbounded function Microscopic Approach Universal Sequences diamond star Commutative cancellative semigroups Large Cardinals Antichain Strongly Luzin set PFA OCA Small forcing Amenable C-sequence Almost Souslin b-scale Reduced Power Martin's Axiom O-space Diamond for trees Ulam matrix square principles Shelah's Strong Hypothesis Vanishing levels Whitehead Problem Precaliber Generalized descriptive set theory specializable Souslin tree free Souslin tree Erdos-Hajnal graphs Respecting tree Rado's conjecture approachability ideal weak diamond Singular cofinality Parameterized proxy principle 54G20 Fodor-type reflection SNR Foundations Coherent tree Rock n' Roll Souslin Tree Subnormal ideal
Tag Archives: Rainbow sets
Prolific Souslin trees
In a paper from 1971, Erdos and Hajnal asked whether (assuming CH) every coloring witnessing $\aleph_1\nrightarrow[\aleph_1]^2_3$ has a rainbow triangle. The negative solution was given in a 1975 paper by Shelah, and the proof and relevant definitions may be found … Continue reading
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Tagged Rainbow sets, Souslin Tree, Square-Brackets Partition Relations
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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
Comparing rectangles with squares through rainbow sets
In Todorcevic’s class last week, he proved all the results of Chapter 8 from his Walks on Ordinals book, up to (and including) Theorem 8.1.11. The upshots are as follows: Every regular infinite cardinal $\theta$ admits a naturally defined function … Continue reading
