Transversals in Superdisjoint T(3)-families of Translates

Line Transversals to Translates of Unit Discs

Let F be a family of convex figures in the plane. We say that F has property T if there exist a line intersecting every member of F . Also, we say that the family F has property T (k) if every k−membered subfamily of F has property T . In this talk we will give some results concerning families of translates of the unit disc which have property T (3) or T (4).

Disjoint Common Partial Transversals of Two Families of Sets

The triples of geometric permutations for families of disjoint translates

A line meeting a family of pairwise disjoint convex sets induces two permutations of the sets. This pair of permutations is called a geometric permutation. We characterize the possible triples of geometric permutations for a family of disjoint translates in the plane. Together with earlier studies of geometric permutations this provides a complete characterization of realizable geometric permut...

Intersections of Multiplicative Translates of 3-Adic Cantor Sets II: Two Infinite Families

Transversals in Trees

The n− 2 upper bounds in (1) are attained simultaneously if and only if T is the path (the tree with precisely one leaf); the n− 2 lower bounds in (1) are attained simultaneously if and only if T is the star (the tree where all leaves are children of the root). Jonathan David Farley asked how high can these lower bounds be raised if each node of T has at most two children; he offered a creative...