Monochromatic loop in plane Suppose all the points in the plane are coloured with two colours. Are we guaranteed to find a continuous closed monochromatic path in the plane ? I believe the answer is yes, and then what if countable infinite colours are used ? If the answer is no, then what if it's just a path (not necessarily a loop) ?

No. Colour all points $(x,y)$ with $x$ rational red, all points with $x$ irrational blue. Then any monochromatic path is contained in a vertical line.

Using the Axiom of Choice, we can get a colouring with no monochromatic paths at all. The set of paths in the plane has cardinality $c$. Index them by the first ordinal of cardinality $c$. Then using transfinite induction, we can produce disjoint sets $A$ and $B$ such that every path contains at least one member of $A$ and at least one member of $B$. Colour $A$ red and $B$ blue; it doesn't matter what you do to $(A \cup B)^c$.