TITLE: Let $S\subset [0,1]^2$ be uncountable. Is there a line which contains infinitely many points of $S$?
QUESTION [5 upvotes]: If not, how many points can be guaranteed?
Also, I'm not sure about my tag. This is a pretty general question. I figured General Topology is close.
EDIT:
Someone paraphrased this nicely. "Let $S\subset [0,1]^2$ be uncountable. Is there a line which contains infinitely many points of $S$?"

REPLY [22 votes]: Hint: Think about a cirlce $S$ inside $[0,1]\times [0,1].$

REPLY [3 votes]: There is a subset $S$ of size $\omega_1$ of the unit square such that any line contains at most two points of $S$.  Start by picking two points of the square.  Draw the line through them.  Pick a third point, avoiding the line you drew.  Draw the lines from the new point through all the previous points.  Pick the next point avoiding all the lines you drew.  Continue by transfinite induction until you have $\omega_1$ points.  At all stages of the construction you have only used a countable number of points, so there are a countable number of lines drawn.  As Lebesgue measure is countably additive, the area of all the lines is zero so there are points available to continue.