Geometric graphs: the crossing lemma.
Application of crossing lemma to Erdos problems:
Geometric Incidences, Repeated distance problem, distinct distances problem. Selection lemmas. Counting $k$-sets. An application of incidences to additive number theory.

Coloring and hiting problems for geometric hypergraphs : $VC$-dimension, Transversals and Epsilon-nets. Weak eps-nets for convex sets. $(p,q)$-Theorem, Conflict-free colorings.

Arrangements : Davenport Schinzel sequences and sub structures in arrangements.

Geometric Ramsey and Turan type theorems: Application of Dilworth theorem, Erdos-Szekeres theorem for convex sets, quasi-planar graphs.