In what follows, I will use the following acronyms:

$AS$:= Freiling's Axiom of Symmetry

$LM$:="Every set of reals is Lebesgue measurable."

$WCH$:="every uncountable subset of $\mathbf R$ can be put into 1-1 correspondence with $\mathbf R$."

$DC$:= Axiom of dependent choice

In his FOM blogpost, August 3, 2011 (01:16:58), to Timothy Chow, Ali Enayat stated that

"...$ZF$+$AS$+$LM$+$WCH$ holds in Solovay's model since in Solovays model every uncountable subset of $\mathbf R$ has a perfect subset."

Suppose one would replace $AS$ with

($A_{\lt2^{\aleph_0}}$) $\forall$$f$:$\mathbf R$$\rightarrow$$\mathbf R_{\lt 2^{\aleph_0}}$ ($\exists$$x_1$, $x_2$)($x_2$$\notin$$f$($x_1$) $\land$ $x_1$$\notin$$f$($x_2$), where $\mathbf R_{\lt2^{\aleph_0}}$ is the set of all subsets of $\mathbf R$ of cardinality $\lt$$2^{\aleph_0}$,

Does $ZF$+$A_{\lt2^{\aleph_0}}$+$LM$ imply $\lnot$$WCH$?

If not, what is the current state of research in discovering models of $ZF$ in which $LM$+$\lnot$$WCH$ holds?

Consider the following variant of Freiling's Axiom of Symmetry, $\mathsf{AS}$, which will be denoted $A_{< 2^{\aleph_0}}$:

• given any function $f$ from $\mathbb{R}$ into the families of of subsets of $\mathbb{R}$ of size $< 2^{\aleph_0}$ there are $x_1,x_2 \in \mathbb{R}$ such that $x_1 \notin f(x_2)$ and $x_2 \notin f(x_1)$.

Does $\mathsf{ZF} + A_{<2^{\aleph_0}} + \mathsf{LM}$ imply $\neg \mathsf{WCH}$? If not, what is the current state of research in discovering models of $\mathsf{ZF}$ in which $\mathsf{LM} +\lnot \mathsf{WCH}$ holds?

Here

• $\mathsf{LM}$ is the statement that every set of reals is Lebesgue measurable,
• $\mathsf{WCH}$ is the statement that every uncountable subset of $\mathbb R$ can be put into 1-1 correspondence with $\mathbb R$.

For background to this question, in his 3 August 2011 FOM post to Timothy Chow, Ali Enayat stated that

$\mathsf{ZF} + \mathsf{AS} + \mathsf{LM} + \mathsf{WCH}$ holds in Solovay's model since in Solovay's model every uncountable subset of $\mathbb{R}$ has a perfect subset.