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

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