Regularity lemma: For all eps exists K such that for all k>K, we can partition the vertices of any graph into k groups evenly
such that (except at most an eps fraction of all) group pairs behave like a random graph of the respective density,
that is the number of edges between any two subsets of two groups deviates from the expected number by O(eps (n/k)^2).
We only give the main idea of the proof for this theorem (for the proof of a weaker hypergraph version, see Matousek):
Fix a graph G with n vertices and denote by B_k=(P_1,..,P_k) a partition of its vertices.
Between any two partitions, i,j, denote the edge density by d(i,j).
For any two vertices, u from P_i and v from P_j, let g(u,v)=d(i,j)-E(u,v).
Define the energy of B_k as E_k=1/n^2 sum_{u,v} g^2(u,v).
Note that 0<=E_k<=1 and if B_l is a refinement of B_k, then E_l<=E_k.
Ex: Prove the above. (Sol: \sum_i (a_i-x)^2 attains its min at x=average(a_i))
Start with B_1=(V) and refine it always by dividing one partition into two, obtaining maximum energy decrease.
Once the energy will increase by less than eps.
Moreover, for a long time the total increase will be less than eps.
But if E_k - E_{k+k^2}<eps, then there can be at most eps k^2 pairs that are not eps-regular in B_k, done.
Transversal number: tau(F), packing number: nu(F)
Konig-Hall thm: tau(E)=nu(E) for bipartite graphs.
Fractional versions are equal (for finite sets), follows from LP duality.
Lovasz (without proof): tau(F)<= tau*(F)(1+ln maxdeg(F)).
Hw: Given a tournament, let F={Dv: v\in V}, where Dv={u: vu\in E or u=v}.
a) How much is nu(F)?
b) Give an example where tau(F)>100.
c) Prove tau*(F)<2.
Epsilon net for finite sets and for measures.
Thm: Exists 1/r-net of size Cdr log r, if VC-dim(F)<=d, where C<20 and C->1 as r-> infty.
We give proof next week and def of VC-dim at end of class.
Cor: tau(F)<=Cdtau*(F) log(tau*(F)).
Proof: Define probability distribution concentrated on points of tau*, this makes every set 1/tau*-big.
VC-dim: Size of biggest shattered point set.
It can also be infinite, e.g. for convex sets of plane.
Hw: Determine VC-dim of halfspaces in d-dim.