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Category Archives: Combinatorics
Analysis of Boolean functions – week 2
Post on week 1; home page of the course analysis of Boolean functions Lecture II: We discussed two important examples that were introduced by BenOr and Linial: Recursive majority and tribes. Recursive majority (RM): is a Boolean function with variables … Continue reading
Posted in Combinatorics, Computer Science and Optimization, Probability, Teaching
Tagged Boolean functions, Tribes
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Around Borsuk’s Conjecture 3: How to Save Borsuk’s conjecture
Borsuk asked in 1933 if every bounded set K of diameter 1 in can be covered by d+1 sets of smaller diameter. A positive answer was referred to as the “Borsuk Conjecture,” and it was disproved by Jeff Kahn and me in 1993. … Continue reading
Analysis of Boolean Functions – week 1
Home page of the course. In the first lecture I defined the discrete ndimensional cube and Boolean functions. Then I moved to discuss five problems in extremal combinatorics dealing with intersecting families of sets. 1) The largest possible intersecting family … Continue reading
Poznań: Random Structures and Algorithms 2013
Michal Karonski (left) who built Poland’s probabilistic combinatorics group at Poznań, and a sculpture honoring the Polish mathematicians who first broke the Enigma machine (right, with David Conlon, picture taken by Jacob Fox). I am visiting now Poznań for the 16th … Continue reading
Posted in Combinatorics, Conferences, Open problems, Philosophy, Probability
Tagged Poznan, RSA
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LawlerKozdronRichardsStroock’s combined Proof for the MatrixTree theorem and Wilson’s Theorem
David Wilson and a cover of Shlomo’s recent book “Curvature in mathematics and physics” A few weeks ago, in David Kazhdan’s basic notion seminar, Shlomo Sternberg gave a lovely presentation Kirchhoff and Wilson via Kozdron and Stroock. The lecture is based on … Continue reading
Posted in Combinatorics, Computer Science and Optimization, Probability
Tagged David Wilson, Gustav Kirchhoff, Trees
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Some old and new problems in combinatorics and geometry
Paul Erdős in Jerusalem, 1933 1993 I just came back from a great Erdős Centennial conference in wonderful Budapest. I gave a lecture on old and new problems (mainly) in combinatorics and geometry (here are the slides), where I presented twenty … Continue reading
Andriy Bondarenko Showed that Borsuk’s Conjecture is False for Dimensions Greater Than 65!
The news in brief Andriy V. Bondarenko proved in his remarkable paper The Borsuk Conjecture for twodistance sets that the Borsuk’s conjecture is false for all dimensions greater than 65. This is a substantial improvement of the earlier record (all dimensions … Continue reading
New Ramanujan Graphs!
Margulis’ paper Ramanujan graphs were constructed independently by Margulis and by Lubotzky, Philips and Sarnak (who also coined the name). The picture above shows Margulis’ paper where the graphs are defined and their girth is studied. (I will come back to the question … Continue reading
Posted in Algebra and Number Theory, Combinatorics, Open problems
Tagged Ramanujan graphs
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Andrei
Andrei Zelevinsky passed away a week ago on April 10, 2013, shortly after turning sixty. Andrei was a great mathematician and a great person. I first met him in a combinatorics conference in Stockholm 1989. This was the first major … Continue reading
Test Your Intuition (19): The Advantage of the Proposers in the Stable Matching Algorithm
Stable mariage The GaleShapley stable matching theorem and the algorithm. GALESHAPLEY THEOREM Consider a society of n men and n women and suppose that every man [and every woman] have a preference (linear) relation on the women [men] he [she] knows. Then … Continue reading