Let T be a finite set of n open triangles in R2, and 1 < k < n be an integer. We say that a point in R2 is k-deep (with respect to T) if it pierces k triangles of T. n n Show that there is a subset T' C T of cardinality 0 (—k log —k)' and whose union covers every k-deep point in R2. Hint: The e-net theorem.

Let T denote the family of all the open triangles in R2. Consider the so called dual range space (T,R2) which encompasses all the hypergraphs (T, E) whose (finite) vertex set satisfies T C T, and each hyper-edge in E is of the form ; := {t E T I p e tl. 1 Show that every hypergraph (T, E) as above in the dual range space (T, R2) has at most 0 (1T12) hyperedges.

• Let T be a finite set of n open triangles in R2, and 1 < k < n be an integer. We say that a point in R2 is k-deep (with respect to T) if it pierces k triangles of T. n n Show that there is a subset T’ C T of cardinality 0 (—k log —k)’ and whose union
covers every k-deep point in R2. Hint: The e-net theorem.