The Number of Arcs in $${\mathbb {F}}_q^2$$ of a Given Cardinality

Abstract A subset of $${\mathbb {F}}_q^2$$ is called an arc if it does not contain three collinear points. We show that there are at most $$\left( {\begin{subarray}{c}(1 + o(1))q\\ m\end{subarray}}\right) $$ arcs of size $$m \gg q^{1/2} (\log q)^{3/2}$$ , nearly matching a trivial lower bound of $$\left( {\begin{subarray}{c}q\\ m\end{subarray}}\right) $$ . This was previously known to hold for $$m \gg q^{2/3} (\log q)^3$$ , by a result of Bhowmick and Roche-Newton. The lower bound on m is best possible up to a logarithmic factor.