A critical ideal with respect to Mazurkiewicz's theorem in terms of the Katetov order

Consider a property of ideals: property(I). One can ask whether there
is a critical ideal C for property(I) in terms of the Katetov order
<_K i.e. property(I) <=> C <_K I. Many examples of critical ideals for
various properties can be found in Hrusak's paper ``Combinatorics of
filters and ideals''.
In the first part of my talk I show a few examples of properties of
ideals and critical ideals for these properties in the Katetov order.
In the second part I show a critical ideal with respect to
Mazurkiewicz's theorem. At the end I show a relationship between Part
2 and a question of Hrusak about extendability of ideals to F_sigma
ideals.
A theorem of Mazurkiewicz I'm going to talk about says that for every
bounded sequence of continuous functions f_n:R->R there is a perfect
set P such that the sequence (f_n|P) has a pointwise convergent
subsequence.