Counting Arcs in \({\mathbb {F}}_q^2\)

An arc in \(\mathbb F_q^2\) ${\mathbb{F}}_q^2$ is a set \(P \subset \mathbb F_q^2\) $P\subset {\mathbb{F}}_q^2$ such that no three points of P are collinear. We use the method of hypergraph containers to prove several counting results for arcs. Let \({\mathcal {A}}(q)\) $\mathcal{A}(q)$ denote the family of all arcs in \(\mathbb F_q^2\) ${\mathbb{F}}_q^2$ . Our main result is the bound \(\begin{aligned} |{\mathcal {A}}(q)| \le 2^{(1+o(1))q}. \end{aligned}\) $$\begin{array}{c}\hfill |\mathcal{A}(q)|\le 2^{(1+o(1))q}.\end{array}$$ This matches, up to the factor hidden in the o(1) notation, the trivial lower bound that comes from considering all subsets of an arc of size q. We also give upper bounds for the number of arcs of a fixed (large) size. Let \(k \ge q^{2/3}(\log q)^3\) $k\ge q^{2/3}{(logq)}^3$ , and let \({\mathcal {A}}(q,k)\) $\mathcal{A}(q,k)$ denote the family of all arcs in \(\mathbb F_q^2\) ${\mathbb{F}}_q^2$ with cardinality k. We prove that \(\begin{aligned} |{\mathcal {A}}(q,k)| \le \left( {\begin{array}{c}(1+o(1))q\\ k\end{array}}\right) . \end{aligned}\) $$\begin{array}{c}\hfill |\mathcal{A}(q,k)|\le \left(\begin{array}{c}(1+o(1))q\\ k\end{array}\right).\end{array}$$ This result improves a bound of Roche-Newton and Warren [12]. A nearly matching lower bound \(\begin{aligned} |{\mathcal {A}}(q,k)| \ge \left( {\begin{array}{c}q\\ k\end{array}}\right) \end{aligned}\) $$\begin{array}{c}\hfill |\mathcal{A}(q,k)|\ge \left(\begin{array}{c}q\\ k\end{array}\right)\end{array}$$ follows by considering all subsets of size k of an arc of size q.