We will present classical multidimensional transfinite
inductive arguments leading under the continnum hypothesis to a strong
Luzin set (Todorcevic) and to a c.c.c. partial order whose square is not
c.c.c. (Galvin). Both of these constructions are related to some
colorings of pairs of countable ordinals which found applications in
algebra, analysis and topology (e.g., see Wednesday seminar this week).
If time permits, we will discuss what happens after adding one or many
Cohen reals.