# Tag Archives: Extremal combinatorics

## Extremal Combinatorics V: POSETS

This is the remaining post V on partially ordered sets of my series on extremal combinatorics (I,II,III,IV,VI).  We will talk here about POSETS – partially ordered sets. The study of order is very important in many areas of mathematics starting … Continue reading

## Analysis of Boolean Functions – week 1

Home page of the course. In the first lecture I defined the discrete n-dimensional 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

## Cap Sets, Sunflowers, and Matrix Multiplication

This post follows a recent paper On sunflowers  and matrix multiplication by Noga Alon, Amir Spilka, and Christopher Umens (ASU11) which rely on an earlier paper Group-theoretic algorithms for matrix multiplication, by Henry Cohn, Robert Kleinberg, Balasz Szegedy, and Christopher Umans (CKSU05), … Continue reading

## Extremal Combinatorics on Permutations

We talked about extremal problems for set systems: collections of subsets of an element sets, – Sperner’s theorem, the Erdos-Ko-Rado theorem, and quite a few more. (See here, here and here.) What happens when we consider collections of permutations rather than … Continue reading

Posted in Combinatorics | | 9 Comments

## Frankl-Rodl’s Theorem and Variations on the Cap Set Problem: A Recent Research Project with Roy Meshulam (A)

Voita Rodl I would like to tell you about a research project in progress with Roy Meshulam. (We started it in the summer, but then moved to other things;  so far there are interesting insights, and perhaps problems, but not substantial … Continue reading

Posted in Combinatorics, Open problems | | 11 Comments

## Lovasz’s Two Families Theorem

Laci and Kati This is the first of a few posts which are spin-offs of the extremal combinatorics series, especially of part III. Here we talk about Lovasz’s geometric two families theorem.     1. Lovasz’s two families theorem Here … Continue reading

Posted in Combinatorics, Convexity, Open problems | | 7 Comments

## Extremal Combinatorics IV: Shifting

Compression   We describe now a nice proof technique called “shifting” or “compression” and mention a few more problems.   The Sauer-Shelah Lemma: Let . Recall that a family shatters a set if for every there is   such that … Continue reading

Posted in Combinatorics, Open problems | | 15 Comments

## Extremal Combinatorics III: Some Basic Theorems

. Shattering Let us return to extremal problems for families of sets and describe several basic theorems and basic open problems. In the next part we will discuss a nice proof technique called “shifting” or “compression.” The Sauer-Shelah (-Perles -Vapnik-Chervonenkis) Lemma: (Here we write .) … Continue reading

Posted in Combinatorics, Open problems | | 17 Comments

## Extremal Combinatorics II: Some Geometry and Number Theory

Extremal problems in additive number theory Our first lecture dealt with extremal problems for families of sets. In this lecture we will consider extremal problems for sets of real numbers, and for geometric configurations in planar Euclidean geometry. Problem I: Given a set A of … Continue reading

## Local Events, Turan’s Problem and Limits of Graphs and Hypergraphs

I will write a little about how hectic things are now here at HU, and make two (somewhat related) follow-ups on previous posts: Tell you about Turan’s problem, and about Balázs Szegedi’s lecture from Marburg dealing with limits of graphs and hypergraphs.  Local Events … Continue reading

Posted in Combinatorics, Open problems | | 4 Comments