A subset $A$ of the real line is called a Q-set if any subset of of $A$ is of type $F_\sigma$ in $A$.
Let $\mathfrak q_0$ be the smallest cardinality of a subset $X\subset\mathbb R$ which is not a Q-set in $\mathbb R$.
It can be shown that $q_0$ is the smallest cardinality of a set $A\subset \mathbb R$ containing two disjoint subsets $A_1,A_2$ that cannot be separated by $F_\sigma$-subsets of $\mathbb R$.
We say that two subsets $A,B\subset\mathbb R$ can be separated by $F_\sigma$-sets if there are two disjoint $F_\sigma$-sets $\tilde A$ and $\tilde B$ in $\mathbb R$ such that $A\subset\tilde A$ and $B\subset\tilde B$.
Taking into account the characterization of $\mathfrak q_0$ in terms of $F_\sigma$-separation, let us define two modifications of $\mathfrak q_0$.
Let $\mathfrak q_1$ be the smallest cardinal $\kappa$ for which there exists a set $A\subset \mathbb R$ of cardinality $|A|\le\kappa$ and a family $\mathcal B$ of compact subsets of $\mathbb R$ such that $|\mathcal B|\le\kappa$, $A\cap\bigcup\mathcal B=\emptyset$ and the sets $A$ and $\bigcup\mathcal B$ cannot be separated by $F_\sigma$-subset of $\mathbb R$.
Let $\mathfrak q_2$ be the smallest cardinal $\kappa$ for which there exists two families $\mathcal A$ and $\mathcal B$ of compact subsets of $\mathbb R$ such that $\max\{|\mathcal A|,|\mathcal B|\}\le\kappa$, the set $\bigcup\mathcal A$ and $\bigcup\mathcal B$ are disjoint but cannot be separated by $F_\sigma$-subsets of $\mathbb R$.
It is clear that $\mathfrak q_2\le \mathfrak q_1\le \mathfrak q_0$. By analogy with the proof of Theorem 2 in this paper, it can be shown that $\mathfrak p\le\mathfrak q_2$. So, we have the inequalities $$\mathfrak p\le\mathfrak q_2\le\mathfrak q_1\le\mathfrak q_0.$$ The cardinal $\mathfrak q_0$ has been studied in the literature. What about the 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 $\mathfrak q_1<\mathfrak q_0$ consistent?
