Counting Arcs in F q 2 .

An arc in F q 2 is a set P ⊂ 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 A ( q ) denote the family of all arcs in F q 2 . Our main result is the bound | A ( q ) | ≤ 2 ( 1 + o ( 1 ) ) q . 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 ≥ q 2 / 3 ( log q ) 3 , and let A ( q , k ) denote the family of all arcs in F q 2 with cardinality k . We prove that | A ( q , k ) | ≤ ( 1 + o ( 1 ) ) q k . This result improves a bound of Roche-Newton and Warren [12]. A nearly matching lower bound | A ( q , k ) | ≥ q k follows by considering all subsets of size k of an arc of size q .
(© The Author(s) 2024.)