$F_{\sigma }$-ideals, colorings, and representation in Banach spaces

In recent works by L. Drewnowski and I. Labuda and J. Martínez 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 $d\ge 3$, 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.