A subset $X\subset \mathbb R$ is called
$\bullet$ a $Q$-set if for every subset $A\subset X$ there exists a $\sigma$-compact set $C\subset\mathbb R$ such that $C\cap X=A$;
$\bullet$ a strong $Q$-set if for every Polish subspace $P\subset \mathbb R$ and every subset $A\subset X\cap P$ there exists a $\sigma$-compact set $C\subset P$ such that $C\cap X=A$;
$\bullet$ a universal $Q$-set if for every continuous injective map $f:P\to \mathbb R$ form a Polish space $P$ and every subset $A\subset f^{-1}(X)$ there exists a $\sigma$-compact set $C\subset P$ such that $C\cap f^{-1}(X)=A$.
It is clear that $$\mbox{countable $\Rightarrow$ universal $Q$-set $\Rightarrow$ strong $Q$-set $\Rightarrow$ $Q$-set}.$$
It can be shown that every $Q$-set $X\subset\mathbb R$ of cardinality $X\le \mathfrak b$ is a universal $Q$-set.
Let $\mathfrak q_0$ be the smallest cardinality $|X|$ of a subset $X\subset \mathbb R$ which is not a $Q$-set. It is known that $$\mathfrak p\le\mathfrak q_0\le \min\{\mathfrak b,\mathrm{non}(\mathcal N)\}.$$
Using the inequality $\mathfrak q_0\le\mathfrak b$, it can be shown that every subset $X\subset\mathbb R$ of cardinality $|X|<\mathfrak q_0$ is a universal $Q$-set. So, $\mathfrak q_0$ coincides with the smallest cardinality of a subset $X\subset \mathbb R$ which fails to be a strong (or universal) $Q$-set.
Let $\mathfrak q$ (resp. $\mathfrak{q_s}$, $\mathfrak q_u$) be the smallest cardinal $\kappa$ such that every subset $X\subset\mathbb R$ of cardinality $|X|\ge\kappa$ is not a (resp. strong, universal) $Q$-set. It can be shown that $$\mathfrak q_0\le \min\{\mathfrak b,\mathfrak q\}\le\mathfrak q_u\le\mathfrak q_s\le \mathfrak q\le\log(\mathfrak c^+),$$where $\log(\mathfrak c^+):=\min\{\kappa:2^\kappa>\mathfrak c\}$. So, under $2^{\omega_1}>\mathfrak c$, $\mathfrak q=\omega_1$, which means that every $Q$-set is countable.
The consistency of the strict inequality $\mathfrak q_0=\mathfrak b<\mathfrak q$ was proved by Judah and Shelah.
Problem. Which of the strict inequalities $\mathfrak q_0<\mathfrak q_u$, $\mathfrak q_u<\mathfrak q_s$, $\mathfrak q_s<\mathfrak q$ are contistent?