  An arc is a subset of $\mathbb F_q^2$ which does not contain any collinear
triples. Let $A(q,k)$ denote the number of arcs in $\mathbb F_q^2$ with
cardinality $k$. This paper is primarily concerned with estimating the size of
$A(q,k)$ when $k$ is relatively large, namely $k=q^t$ for some $t>0$. Trivial
estimates tell us that
  \[
  {q \choose k} \leq A(q,k) \leq {q^2 \choose k}.
  \]
  We show that the behaviour of $A(q,k)$ changes significantly close to
$t=1/2$. Below this threshold an elementary argument is used to prove that the
trivial upper bound above cannot be improved significantly.
  On the other hand, for $t \geq 1/2+\delta$, we use the theory of hypergraph
containers to get an improved upper bound
  \[
  A(q,k) \leq {q^{2-t+2\delta} \choose k}.
  \]
  This technique is also used to give an upper bound for the size of the
largest arc in a random subset of $\mathbb F_q^2$ which holds with high
probability. For example, we prove that a $p$-random subset $Q \subset \mathbb
F_q^2$ with $q^{-3/2}<p<q^{-1}$ contains an arc of size $\Omega(q^{1/2})$ with
high probability. The result is optimal for this range of $p$.
  Finally, this optimal bound for arcs in random sets is used to prove a finite
field analogue of a result of Balogh and Solymosi, with a better exponent:
there exists a subset $P \subset \mathbb F_q^2$ which does not contain any
collinear quadruples, but with the property that for every $P' \subset P$ with
$|P'| \geq |P|^{3/4+o(1)}$, $P'$ contains a collinear triple.
