I need to prove the following:
Let $S$ be a set of $n$ points in the plane, and let t be the number of lines that pass through exactly $√n$ points of S. Prove that $t = O( √ n)$.
I am pretty new to combinatorics.
Why wouldn't be answer simply $t = √ n$?
In the worst case, Were gonna need a line for each of the $√ n$ Points.
To answer your question (not proving the $O(\sqrt{n})$ itself, it is "why not $t=\sqrt{n}$" or even "why not $t\le\sqrt{n}$"), there are a lot of examples that $t\le\sqrt{n}$ won't hold: think about a $t\times t$ square grid, and there are $t$ horizonal lines and $t$ vertical lines, and thus $2t=2\sqrt{n}$ lines.
However, as @AHusain said, $t\le 3\sqrt{n}$, is also not correct, too. Think about the counterexample: nine points $(-2,1),(0,1),(2,1),(-1,0),(0,0),(1,0),(-2,-1),(0,-1),(2,-1)$ and there are ten such lines: $y=1,y=0,y=-1,x=0,x=2y,x=-2y,y=x+1,y=x-1,y=-x+1,y=-x-1$.
(Maybe this answer is better to be a comment, but I don't have enough reputation...)