Abstract
The continuum function is the function which assigns to every infinite cardinal κ the value 2κ. Easton showed that ZFC puts very few restrictions on the continuum function on regular cardinals.
If we study the continuum function not over ZFC, but over ZFC + large cardinals, the situation changes, and new and interesting restrictions start to appear. It is known that many properties of large cardinals can be reasonably formulated for small cardinals, even below ℵω: these properties are often called "traces" of large cardinals.
If one removes the property of inaccessibility from the definition of a weakly compact cardinal, one ends up with the property that every κ-tree for the given regular cardinal κ has a cofinal branch - we say that κ has the tree property. In this talk we focus on the restrictions imposed on the continuum function by the tree property.