Combinatorics
Professor Imre Leader
2. Chapter II: Isopermietric inequalities
How tightly can we pack a set of given size?
For example, among subsets of R2 of given area, the disc has the smallest perimeter. Similarly
in R3 .
Im supposed to ramble about s

Combinatorics
Professor Imre Leader
Course overview
This course will be organized in three chapters:
1. Set systems: a simple combinatorial setting (subsets of the powerset of a finite set) where
interesting structure emerges;
2. Isoperimetric inequalitie

Combinatorics: Problem set 2
1. Problem 1
Observe that if i(A) indicates the size of the largest intersecting family contained in a set
system A, we have
X
i(X (r) )
i(X (j) )
jr
(r)
because if A X
is intersecting, so is every slice A X (j) . But the Erd

Combinatorics: Problem set 1
Professor Imre Leader
1. Problem 4
Remember the matchings we constructed in class between X (r) and X (r+1) by Halls theorem!
We can use these here to push our set system to the middle two layers.
More formally, among all set

Combinatorics
Professor Imre Leader
3. Chapter III: intersecting families
Say A P(X) is t-intersecting if |x y| t for all x, y A. How large can a t-intersecting
family be?
n
For t = 2, could take x 1, 2 x of size 24 . But! We can beat it by taking all