I'm trying to pinpoint the "intuitive argument" for Freiling's Axiom of Symmetry. It's meant to be a "probabilistic" argument, so thinking about what seems to me to be the probabilistic intuition, it seems natural to ask:

Is it a theorem of ZFC that there exists $c>0$ and a set $S \subset [0,1] \times [0,1]$ such that

• for every $x \in [0,1]$, $\{y \in [0,1] : (x,y) \in S\}$ is contained in a Borel set of zero Lebesgue measure;
• for every $y \in [0,1]$, $\{x \in [0,1] : (x,y) \in S\}$ contains a Borel set of Lebesgue measure greater than or equal to $c$?

(If the answer is yes, then the natural follow up question is whether we can take $c=1$; but this seems less important.)