We study \(B\)- and \(C\)-ideals associated with sequences in Banach spaces, where \(C((x_n)n)\) consists of sets for which the series \(\sum_{n\in A} x_n\) is unconditionally convergent, and \(B((x_n)_n)\) consists of sets where the series is weakly unconditionally convergent. We describe these ideals in universal function spaces, particularly in \(C([0,1])\) and \(C(2^{\mathbb N})\), addressing a question by Borodulin-Nadzieja et al.
A key aspect is the role of \(c_0\)-saturated spaces and their connection to c-coloring ideals, which exhibit a rich combinatorial structure. We show that for \(d\ge 3\), the random d-homogeneous ideal is pathological, construct hereditarily non-pathological c-coloring ideals, and prove that every \(B\)-ideal in \(C(K)\), for countable \(K\), contains a c-coloring ideal.
These results highlight the interplay between combinatorial properties of ideals and their Banach space representations. This is a joint work with V. Olmos and C. Uzcátegui.