Sad news: Jim Baumgartner passed away. See here.
Archives
Keywords
Uniformly homogeneous Almost Souslin HOD diamond star Absoluteness Kurepa Hypothesis Diamond Weakly compact cardinal Prikry-type forcing GMA Monotonically far weak square Successor of Regular Cardinal Strongly Luzin set Jonsson cardinal Fast club sap P-Ideal Dichotomy Hedetniemi's conjecture polarized partition relation Erdos Cardinal Slim tree Forcing stationary hitting Interval topology on trees Uniformly coherent approachability ideal Subtle cardinal Foundations Cardinal function Coherent tree Precaliber Parameterized proxy principle L-space Greatly Mahlo SNR Sigma-Prikry Distributive tree Dowker space Singular cofinality Aronszajn tree ccc perfectly normal Poset Ostaszewski square PFA(S)[S] Strongly compact cardinal OCA middle diamond Generalized descriptive set theory Nonspecial tree Respecting tree transformations free Boolean algebra Was Ulam right? Small forcing Hindman's Theorem Knaster Subnormal ideal Closed coloring Hereditarily Lindelöf space Rainbow sets Iterated forcing Sierpinski's onto mapping principle weak Kurepa tree strongly bounded groups Open Access Ascending path Knaster and friends AIM forcing Rock n' Roll super-Souslin tree Fodor-type reflection Prevalent singular cardinals Cardinal Invariants Successor of Singular Cardinal Microscopic Approach Almost countably chromatic Well-behaved magma Universal Sequences Ulam matrix coloring number Strong coloring Almost-disjoint family Postprocessing function specializable Souslin tree Whitehead Problem ZFC construction xbox Chromatic number stick nonmeager set Countryman line Erdos-Hajnal graphs Generalized Clubs higher Baire space Fat stationary set free Souslin tree stationary reflection Souslin Tree square principles Local Club Condensation. positive partition relation Forcing with side conditions tensor product graph C-sequence O-space Singular cardinals combinatorics indecomposable filter Uniformization Luzin set Amenable C-sequence Ineffable cardinal Martin's Axiom Reflecting stationary set Filter reflection Cohen real Shelah's Strong Hypothesis projective Boolean algebra Club Guessing Sakurai's Bell inequality Partition relations for trees Reduced Power very good scale incompactness Ramsey theory over partitions Vanishing levels Axiom R square Partition Relations regressive Souslin tree unbounded function Square-Brackets Partition Relations S-Space Subadditive Diamond-sharp Rado's conjecture b-scale Chang's conjecture Singular Density Large Cardinals Dushnik-Miller club_AD Commutative cancellative semigroups Forcing Axioms Mandelbrot set Constructible Universe Antichain Selective Ultrafilter full tree PFA Ascent Path Subtle tree property Non-saturation 54G20 countably metacompact Diamond for trees reflection principles Entangled linear order weak diamond Minimal Walks Analytic sets Lipschitz reduction Commutative projection system Intersection model
That’s very sad news indeed.
That is sad.
(*) I will always associate Jim with the phrase “Just Baumgartner it!”. Simon Thomas’s word for bringing in an ultrafilter, seemingly for no reason, and letting it guide your decisions.
(**) I would also like to note that Jim shares his first two names with the actor who portrayed Darth Vader.
This is very sad news indeed. The email sent around Dartmouth said that he passed away while his whole family was visiting for the holidays, so Jim was fortunate to spend his last moments surrounded by family and high spirits.
Baumgartner proved that, in contrast to (weak) square, partial square may hold at a singular cardinal (of any cofinality) above a supercomapct carinal. He also proved that after collapsing a weakly comapct cardinal to $omega_2$, every statinoary subset of $S^2_0$ reflects.
These two gems together with an additional argument shows that $square^*_{aleph_{omega_1}}$ may be introduced by a cofinality-preserving small forcing.