Partition Thm[Fox et al]: For all eps exists K such that for all k>K, finite P in gen pos in d-dim and q point
we can partition P into k groups evenly such that
for all but at most an eps fraction of the d+1-tuples i_0,..,i_d, if we take a d-simplex with vertices from i_0,..,i_d,
then either all or none such simplices contain q.
(For proof see paper of Fox, Gromov,...,Pach: Overlap properties of geometric expanders.)
Ex: Prove the theorem in 2-dim.
Note: Instead of finite point set, we could take cont. prob. measure,
instead of containing q, any bounded degree semi-algebraic relation of h vertices.
This is a geometric version of the hypergraph regularity lemma.
Cor[Pach]: For all d exists c such that for any n points P in gen pos in d-dim there is a q such that
we can select d+1 disjoint groups of size cn from P such that any colorful simplex contains q.
Proof: Using first selection lemma, there is a q for which many simplices contain q.
After using partition thm, for a d+1-tuple of indices we must have that they all contain q.
(For proof without partition thm, see Matousek.)
All proof for above theorems rely on the hypergraph analogue of the following important theorem:
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).