# reference request – Two small, uncountable cardinals linked to sets of Q

A subset $$A$$ of the actual line is calling a Q-set if a subset of $$A$$ is of type $$F_ sigma$$ in $$A$$.

Let $$mathfrak q_0$$ to be the smallest cardinality of a subset $$X subset mathbb R$$ which is not a Q-set in $$mathbb R$$.

We can show that $$q_0$$ is the smallest cardinality of a set $$A subset mathbb R$$ containing two disjoint subsets $$A_1, A_2$$ which can not be separated by $$F_ sigma$$-sets of $$mathbb R$$.

We say that two subsets $$A, B subset mathbb R$$ can be separated by $$F_ sigma$$-sets if there are two disjoints $$F_ sigma$$-sets $$tilde A$$ and $$tilde B$$ in $$mathbb R$$ such as $$A subset A tilde$$ and $$B subset tilde B$$.

Taking into account the characterization of $$mathfrak q_0$$ in terms of $$F_ sigma$$-separation, define two modifications of $$mathfrak q_0$$.

Let $$mathfrak q_1$$ to be the smallest cardinal $$kappa$$ for which there is a set $$A subset mathbb R$$ of cardinality $$| A | the kappa$$ and a family $$mathcal B$$ compact subsets of $$mathbb R$$ such as $$| mathcal B | the kappa$$, $$A cap bigcup mathcal B = emptyset$$ and sets $$A$$ and $$bigcup mathcal B$$ can not be separated by $$F_ sigma$$-all set of $$mathbb R$$.

Let $$mathfrak q_2$$ to be the smallest cardinal $$kappa$$ for which there are two families $$mathcal A$$ and $$mathcal B$$ compact subsets of $$mathbb R$$ such as $$max {| mathcal A |, | mathcal B | } the kappa$$, the whole $$bigcup mathcal A$$ and $$bigcup mathcal B$$ are disjoint but can not be separated by $$F_ sigma$$-sets of $$mathbb R$$.

It's clear that $$mathfrak q_2 mathfrak q_1 mathfrak q_0$$. By analogy with the proof of Theorem 2 in this article, we can show that $$mathfrak for mathfrak q_2$$. So, we have the inequalities
$$mathfrak mathfrak q_2 mathfrak q_1 mathfrak q_0.$$
Cardinal $$mathfrak q_0$$ has been studied in the literature. What about cardinals $$mathfrak q_1$$ and $$mathfrak q_2$$?

Problem 1 Have the cardinal characteristics $$mathfrak q_1$$ and $$mathfrak q_2$$ been considered in the literature?

Problem 2 Are the strict inequalities $$mathfrak q_2 < mathfrak q_1$$ and or $$mathfrak q_1 < mathfrak q_0$$ coherent?

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