$F_σ$-ideals, colorings, and representation in Banach spaces

Abstract: In recent works by L. Drewnowski and I. Labuda and J. Mart\'inez et al., non-pathological analytic ( P )-ideals and non-pathological ( F_\sigma )-ideals have been characterized and studied in terms of their representations by a sequence ( (x_n)n ) in a Banach space, as ( \mathcal{C}((x_n)_n) ) and ( \mathcal{B}((x_n)_n) ). The ideal ( \mathcal{C}((x_n)_n) ) consists of sets where the series ( \sum{n \in A} x_n ) is unconditionally convergent, while ( \mathcal{B}((x_n)_n) ) involves weak unconditional convergence. In this paper, we further study these representations and provide effective descriptions of ( \mathcal{B} )- and ( \mathcal{C} )-ideals in the universal spaces ( C([0,1]) ) and ( C(2^{\mathbb{N}}) ), addressing a question posed by Borodulin-Nadzieja et al. A key aspect of our study is the role of the space ( c_0 ) in these representations. We focus particularly on ( \mathcal{B} )-representations in spaces containing many copies of ( c_0 ), such as ( c_0 )-saturated spaces of continuous functions. A central tool in our analysis is the concept of ( c )-coloring ideals, which arise from homogeneous sets of continuous colorings. These ideals, generated by homogeneous sets of 2-colorings, exhibit a rich combinatorial structure. Among our results, we prove that for ( d \geq 3 ), the random ( d )-homogeneous ideal is pathological, we construct hereditarily non-pathological universal ( c )-coloring ideals, and we show that every ( \mathcal{B} )-ideal represented in ( C(K) ), for ( K ) countable, contains a ( c )-coloring ideal. Furthermore, by leveraging ( c )-coloring ideals, we provide examples of ( \mathcal{B} )-ideals that are not ( \mathcal{B} )-representable in ( c_0 ). These findings highlight the interplay between combinatorial properties of ideals and their representations in Banach spaces.