Arcs in Fq2

• Published in: European journal of combinatorics 2022-06, Vol.103, Article 103512
• Main Authors: Roche-Newton, Oliver, Warren, Audie
• Format: Article
• Language: English
• ISSN: 0195-6698; 1095-9971
• DOI: 10.1016/j.ejc.2022.103512

An arc is a subset of Fq2 which does not contain any collinear triples. Let A(q,k) denote the number of arcs in Fq2 with cardinality k. This paper is primarily concerned with estimating the size of A(q,k) when k is relatively large, namely k=qt for some t>0. We show that the behaviour of A(q,k) changes significantly close to t=1/2. On the other hand, for t≥1/2+δ, we use the theory of hypergraph containers to get a non-trivial upper bound A(q,k)≤q2−t+2δk. This technique is also used to give an upper bound for the size of the largest arc in a random subset of Fq2 which holds with high probability. This bound is used to prove a finite field analogue of a result of Balogh and Solymosi (0000), with a better exponent: there exists a subset P⊂Fq2 which does not contain any collinear quadruples, but such that for every P′⊂P with |P′|≥|P|3/4+o(1), P′ contains a collinear triple.