One way of formulating Arhangel'skii's celebrated theorem about the cardinality of Lindelof first-countable spaces is the following (due to Arhangel'skii and Shapirovskii). For every Hausdorff space $X$:
$$|X| \leq 2^{t(X) \cdot \psi(X) \cdot L(X)}$$
where $L(X)$ denotes the Lindelof degree of $X$ (that is, the minimum cardinal $\kappa$ such that every open cover of $X$ has a subcover of cardinality $\leq \kappa$), $\psi(X)$ denotes the pseudocharacter of $X$ (that is, the least cardinal $\kappa$ such that every point in $X$ is a $G_\kappa$ set) and $t(X)$ denotes the tightness of $X$, that is the minimum cardinal $\kappa$ such that for every non-closed set $A \subset X$ and for every point $x \in \overline{A} \setminus A$ there is $C \subset A$ such that $x \in \overline{C}$ and $|C| \leq \kappa$.
The above inequality can be refined by making use of the notion of free sequence, introduced by Arhangel'skii in his original proof of Arhangel'skii's Theorem.
A sequence $\{x_\alpha: \alpha < \kappa\} \subset X$ is said to be free if for every $\beta < \kappa$, $\overline{\{x_\alpha: \alpha < \beta\}} \cap \overline{\{x_\alpha: \alpha \geq \beta \}}=\emptyset$. If we let $F(X)$ be the supremum of cardinalities of free sequences in $X$ it is easy to see that $F(X) \leq L(X) \cdot t(X)$.
Juhasz was the first to note that, for every Hausdorff space $X$:
(*) $$|X| \leq 2^{F(X) \cdot \psi(X) \cdot L(X)}$$
A famous problem in set-theoretic topology asks whether $F(X)$ can be dropped in the above inequality (the answer is "consistently NO" by examples of Shelah, Gorelic, Usuba, Dow, and others...)
In another direction I'd like to ask, can $L(X)$ be dropped from (*)? That is:
QUESTION: Is it true that $|X| \leq 2^{F(X) \cdot \psi(X)}$ for every (regular) space $X$?
I'd expect the answer to be NO, but Juhasz, Soukup and Szentmiklossy have proved the following partial result in the positive direction:
$$|X| \leq 2^{2^{F(X) \cdot \psi(X)}}$$
for every regular space $X$.
As a final side remark note that $\psi(X)$ cannot be dropped from (*). Indeed, the space $X=\sigma(2^\kappa)=\{f \in 2^\kappa: |f^{-1}(1)| < \aleph_0\}$ with the topology induced from $2^\kappa$ is a $\sigma$-compact space of cardinality $\kappa$ with countable tightness (and hence it has countable free sequences).
