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Recent blog posts
- The S-space problem, and the cardinal invariant $\mathfrak b$ April 4, 2013
- An $S$-space from a Cohen real April 3, 2013
- Forcing with a Souslin tree makes $\mathfrak p=\omega_1$ April 1, 2013
- The S-space problem, and the cardinal invariant $\mathfrak p$ March 28, 2013
- Jones’ theorem on the cardinal invariant $\mathfrak p$ March 26, 2013
- Erdős 100 March 26, 2013
- Bell’s theorem on the cardinal invariant $\mathfrak p$ March 21, 2013
- The $\Delta$-system lemma: an elementary proof March 20, 2013
Keywords
Singular cardinals combinatorics Small forcing very good scale stationary reflection Successor of Singular Cardinal P-Ideal Dichotomy projective Boolean algebra b-scale Prikry-type forcing Hereditarily Lindelöf space Almost countably chromatic Diamond Aronszajn tree Chromatic number sap weak square Shelah's Strong Hypothesis Foundations Large Cardinals Square-Brackets Partition Relations polarized partition relation Mandelbrot set Sakurai's Bell inequality Erdos-Hajnal graphs Generalized Clubs Prevalent singular cardinals Ostaszewski square diamond star middle diamond Cardinal function Erdos Cardinal Rock n' Roll reflection principles Souslin Tree Whitehead Problem free Boolean algebra Antichain Axiom R square Dushnik-Miller Club Guessing Minimal Walks S-Space weak diamond Non-saturation approachability ideal Rainbow sets Kurepa Hypothesis Forcing Poset Partition Relations Uniformization Successor of Regular Cardinal Cohen real PFA(S)[S] Singular Density incompactness Rado's conjecture Knaster stationary hitting Singular Cofinality
Tag Archives: 03E02
Rectangular square-bracket operation for successor of regular cardinals
Joint work with Stevo Todorcevic. Extended Abstract: Consider the coloring statement $\lambda^+\nrightarrow[\lambda^+;\lambda^+]^2_{\lambda^+}$ for a given regular cardinal $\lambda$: In 1990, Shelah proved the above for $\lambda>2^{\aleph_0}$; In 1991, Shelah proved the above for $\lambda>\aleph_1$; In 1996, Shelah proved the above … Continue reading
Transforming rectangles into squares, with applications to strong colorings
Abstract: It is proved that every singular cardinal $\lambda$ admits a function $\textbf{rts}:[\lambda^+]^2\rightarrow[\lambda^+]^2$ that transforms rectangles into squares. That is, whenever $A,B$ are cofinal subsets of $\lambda^+$, we have $\textbf{rts}[A\circledast B]\supseteq C\circledast C$, for some cofinal subset $C\subseteq\lambda^+$. As a … Continue reading
