1 Introduction

Over a century ago, Dudeney [8] asked how many points can be placed in an \(n \times n\) grid such that no three are collinear. In 1951 Erdős (published by Roth [13]) showed that when \(n\) is a prime number the set \(\{(i,i^2\) mod \(n)\): \(0\le i < n\}\) contains no collinear triple, whereas a simple upper bound of \(2n\) follows from the pigeonhole principle. Hall et al. [9] subsequently improved Erdős’s lower bound, but despite receiving considerable attention, the problem remains open. In this paper we will consider a closely related problem in the finite field setting.

Let \({\mathbb {F}}_q\) be the finite field of order \(q = p^r\) for some prime p. An arc in \({\mathbb {F}}_q^2\) is a subset of \({\mathbb {F}}_q^2\) with no three points collinear. Let \({\mathcal {A}}(q)\) denote the family of all arcs in \(\mathbb F_q^2\). One of the main goals of this paper is to provide bounds for the cardinality of \({\mathcal {A}}(q)\). For context, observe that the set

$$\begin{aligned} C=\{(x,x^2): x \in \mathbb F_q\} \end{aligned}$$
(1)

is an arc of cardinality q. This is essentially the same as the Erdős construction mentioned above. Since each subset of an arc is also an arc, it immediately follows that

$$\begin{aligned} |{\mathcal {A}}(q)| \ge 2^q. \end{aligned}$$
(2)

We prove the following upper bound, which almost matches (2).

Theorem 1.1

Let q be a prime power which is sufficiently large. There is an absolute constant C such that the set \({\mathcal {A}}(q)\) of all arcs in \(\mathbb F_q^2\) satisfies the bound

$$\begin{aligned} |{\mathcal {A}}(q)| \le 2^{q+Cq^{\frac{4}{5}}(\log q)^2}. \end{aligned}$$

In particular, \(| {\mathcal {A}}(q)| \le 2^{(1+o(1))q}\).

We also consider the set \({\mathcal {A}}(q,k)\) of all arcs of a fixed size k, with a focus on the case when k is large. The problem of bounding the size of \({\mathcal {A}}(q,k)\) was the main focus of a paper of the second author and Warren [12], and this paper is a sequel to [12].

For the case when \(0 \le k < q^{1/2}\), this question was largely settled in [12], where the bounds

$$\begin{aligned} \left( {\begin{array}{c}q^2\\ k\end{array}}\right) e^{\frac{-Ck^3}{q}} \le |{\mathcal {A}}(q,k)| \le \left( {\begin{array}{c}q^2\\ k\end{array}}\right) e^{\frac{-ck^3}{q}} \end{aligned}$$
(3)

were established. In (3), c and C are positive absolute constants. We interpret (3) as a statement that, for small k, a random set of k elements has a fairly high probability of being an arc, with this probability even tending towards 1 for \(k=o(q^{1/3})\).

Write \(k=q^t\). For \(t >1/2\), a significant change of behaviour in terms of the size of \({\mathcal {A}}(q,k)\) was observed in [12]. A better upper bound than that of (3) was established, see [12, Thm. 2]. However, it was not clear whether or not the upper bound given in [12] for this range was optimal.

In this paper, we give improved upper bounds for the size of \({\mathcal {A}}(q,k)\) when k is large.

Theorem 1.2

Let q be a prime power which is sufficiently large and let \(k \ge q^{2/3}(\log q)^3\). Then

$$\begin{aligned} |{\mathcal {A}}(q,k)| \le \left( {\begin{array}{c}(1+o(1))q\\ k\end{array}}\right) . \end{aligned}$$

A more precise version of the statement of Theorem 1.2 which gives information about the value hidden in the o(1) term is given as Theorem 4.2. Theorem 1.2 is close to optimal; the lower bound

$$\begin{aligned} |{\mathcal {A}}(q,k)| \ge \left( {\begin{array}{c}q\\ k\end{array}}\right) \end{aligned}$$

can be seen by considering all subsets of size k of the set C described in (1).

This paper follows a similar approach to that of [12]. In particular, the main tool is the method of hypergraph containers. The theory of hypergraph containers was developed independently by Balogh et al. [2] and Saxton and Thomason [14]. We defer the full statement of the container theorem we use until Sect. 2.1. Roughly speaking, it says that if a hypergraph has a reasonably good edge distribution, we can obtain strong information about where the independent sets of the hypergraph may be found.

In comparison with the predecessor paper [12], there are two main new ideas which allow us to break new ground.

  • To prove Theorem 1.1, we need a supersaturation lemma, namely Corollary 2.3, which gives good bounds for the number of arcs determined by sets with slightly more than q elements, and in particular sets of size \(q+x\) when \(x=o(q)\). See Sect. 2.2 for more background on supersaturation results and their interaction with the method of hypergraph containers.

  • As was the case in [12], we repeatedly apply the hypergraph container theorem to obtain a set of containers for the family of arcs \({\mathcal {A}}(q)\). However, we observe that, as these containers get smaller, we can eventually make use of a non-trivial bound for the maximum co-degree \(\Delta _2\). See Sect. 2.1 for the definition of this parameter. This results in better quantitative information about the set of containers, which in turn yields the improved bound stated in Theorem 1.2.

1.1 Recent Work of Chen, Liu, Nie and Zeng

Since the first version of this paper was published online, work of Chen, Liu, Nie and Zeng [7] has appeared which considers many of the problems studied in this paper and their extensions to the higher dimensional setting. In particular, they prove that there is an absolute constant c such that

$$\begin{aligned} |{\mathcal {A}}(q,k)| \le \left( {\begin{array}{c}cq\\ k\end{array}}\right) , \end{aligned}$$

holds for all integers \(k \ge q^{1/2}(\log q)\). Although the result in [7] is weaker than Theorem 1.2 in the sense that there is a multiplicative constant on the top of the binomial coefficient, it has the significant advantage that it extends to consider almost all possible values of k, thus essentially closing the problem of counting arcs of a fixed size, up to the presence of the constant c. The same paper also closes a gap which was left by our forthcoming Theorem 4.3 concerning the typical size of the largest arc in a random set. The paper [7] uses similar methods to ours, with the augmentation of a so-called “balanced supersaturation" argument allowing for additional progress.

2 Containers and Supersaturation

2.1 Statement of the Container Theorem

The main tool of this paper is a container theorem for 3-uniform hypergraphs. The method of hypergraph containers has had a remarkable impact on extremal combinatorics in recent years (see for example [4] and [10]). This impact is also being felt in Additive Combinatorics (see [1] and [11]) and discrete geometry (see [5]). See [3] for a fairly recent survey of this topic.

Before stating the container theorem to be used, it is necessary to introduce some related quantities. Since we will only apply the container theorem for 3-uniform hypergraphs, we give all of the definitions we need only for this case. A more general form of the statement and definitions we need, adapted to k-uniform hypergraphs, can be found in [5], amongst other places.

For a 3-uniform hypergraph \({\mathcal {H}}=(V,E)\) and \(v \in V\), d(v) denotes the degree of v, i.e. the number of edges which contain v. Let \(d({\mathcal {H}})\) denote the average degree of \({\mathcal {H}}\), so

$$\begin{aligned} d({\mathcal {H}})= \frac{1}{|V|}\sum _{v \in V} d(v)= \frac{3|E|}{|V|}. \end{aligned}$$
(4)

We can also define the co-degree for a subset \(S \subseteq V\) of vertices as

$$\begin{aligned} d(S) = |\{ e \in E({\mathcal {H}}): S \subseteq e \}|. \end{aligned}$$

Using this definition we define the maximum co-degree \(\Delta _2({\mathcal {H}})\) as

$$\begin{aligned} \Delta _2({\mathcal {H}}) = \max _{\begin{array}{c} S \subseteq V \\ |S| = 2 \end{array}}d(S). \end{aligned}$$

More generally, one can define the parameter \(\Delta _t({\mathcal {H}})\), which counts the maximum co-degree among all sets of cardinality t. However, since we only consider 3-uniform hypergraphs in this paper, it turns out that we only use this definition for the case \(t=2\).

For any \(V' \subset V\), \({\mathcal {H}}[V']\) denotes the sub-hypergraph induced by \(V'\).

We now state the container theorem we need, which is a special case of Corollary 3.6 in [14].

Theorem 2.1

Let \({\mathcal {H}}=(V,E)\) be a 3-uniform hypergraph on n vertices, and let \(\epsilon , \tau \in (0,1/2)\). Suppose that

$$\begin{aligned} \frac{\Delta _2({\mathcal {H}})}{d({\mathcal {H}}) \cdot \tau }+ \frac{1}{2d({\mathcal {H}}) \cdot \tau ^{2}} \le \frac{\epsilon }{288}. \end{aligned}$$
(5)

Then there exists a set \({\mathcal {C}}\) of subsets of V such that

  1. i.

    if \(A \subset V\) is an independent set then there exists \(C \in {\mathcal {C}}\) such that \(A \subset C\);

  2. ii.

    \(|E({\mathcal {H}}[C])| \le \epsilon |E({\mathcal {H}})|\) for all \(C \in {\mathcal {C}}\);

  3. iii.

    \(\log |{\mathcal {C}}| \le c n \tau \log (\frac{1}{\epsilon }) \log (\frac{1}{\tau })\),

where c is an absolute constant (we can take \(c= 518400\)).

The set \({\mathcal {C}}\) above is referred to as the set of containers, and a set \(C \in {\mathcal {C}}\) is itself a container.

2.2 Supersaturation Results

In all applications of the method of hypergraph containers, it is necessary to have a supersaturation result. In general terms, a supersaturation lemma is a result that says that, once we have enough elements in our set to guarantee the existence of a certain substructure, we quickly generate many copies of the substructure in question. In our case, this means that we need to show that sets in \(\mathbb F_q^2\) with significantly more than q elements must contain many collinear triples.

Given a set \(P \subset \mathbb F_q^2\), let T(P) denote the number of collinear triples in P. That is,

$$\begin{aligned} T(P)= |\{S \subset P: |S|=3 \text { and the three elements of { S} are collinear}\}|. \end{aligned}$$

It was first proven by Bose [6] that, for q even, any point-set of size \(q+3\) in \(\mathbb {F}_q^2\) contains a collinear triple, and, for q odd, any point-set of size \(q+2\) in \(\mathbb {F}_q^2\) contains a collinear triple. This makes the construction (1) almost best possible. In [12], an application of the Cauchy–Schwarz inequality was used to prove that, for an unspecified absolute constant \(c>0\),

$$\begin{aligned} |P| \ge 4q \Rightarrow T(P) \ge c \frac{|P|^3}{q}. \end{aligned}$$
(6)

This statement is optimal up to the multiplicative constant. This can be seen by taking a p-random subset of \(\mathbb F_q^2\) for some p with order of growth at least 1/q. However, (6) does not say anything about what happens in the range when \(q<|P|<4q\), i.e. when P is only slightly larger than the threshold for guaranteeing at least one arc. In order to prove Theorem 1.1, a good supersaturation result for this range is required. We prove the following rather general supersaturation result for arcs, which includes information about this range.

Lemma 2.2

Let \(k \in \mathbb N \) and let x be an integer satisfying \(0 \le x \le q+1\). Let \(P \subset {\mathbb {F}}_q^2\) with \(|P| = k(q+1)+x+1\). Then

$$\begin{aligned} T(P) \ge \frac{1}{3}\left( \left( {\begin{array}{c}k\\ 2\end{array}}\right) (q+1) + k\cdot x\right) \cdot |P|. \end{aligned}$$

Recall that existence of a collinear triple in a set of size \(q+3\) of \(\mathbb {F}_q^2\) follows from the result of Bose [6]. From Lemma 2.2 we can say a set of size \(q+2+x\) of \(\mathbb {F}_q^2\) for \(x \in \mathbb {N}\) will contain more than \(\frac{qx}{3}\) collinear triples.

Before proving Lemma 2.2, it is convenient to record two corollaries that will be used later.

Corollary 2.3

Let q be a prime power and let \(\frac{4}{q}<\gamma \le 1\). Suppose \(P \subset \mathbb F_q^2\) such that

$$\begin{aligned} T(P) \le \frac{\gamma q^2}{6}. \end{aligned}$$

Then \(|P| \le (1+\gamma )q\).

Proof

We will prove the contrapositive form of the statement; assuming that \(|P| >(1+ \gamma )q\), we prove that \(T(P) > \frac{\gamma q^2}{6}\).

Since \(|P| > (1+ \gamma )q \ge q+2 \), Lemma 2.2 can be applied with \(k=1\), after passing to a subset if necessary. We have

$$\begin{aligned} |P|=q+2+x > (1+ \gamma ) q, \end{aligned}$$

and thus \(x > \gamma q - 2 \ge \frac{\gamma q}{2}\). The latter inequality follows from the assumption that \( \gamma \ge \frac{4}{q}\). Lemma 2.2 then gives

$$\begin{aligned} T(P) \ge \frac{1}{3}x|P| > \frac{1}{3} \frac{\gamma q}{2} |P| \ge \frac{ \gamma q^2}{6} \end{aligned}$$

as required. \(\square \)

Lemma 2.2 can be used to reprove (6). We state the following version, with a concrete multiplicative constant.

Corollary 2.4

Let \(q \ge 8\) be a prime power and let \(P \subset {\mathbb {F}}_q^2\) with \(|P| \ge 2q\). Then

$$\begin{aligned} T(P) \ge \frac{|P|^3}{64q}. \end{aligned}$$

Proof

Write \(|P|=k(q+1)+x+1\) with \(0 \le x \le q+1\). By Lemma 2.2,

$$\begin{aligned} T(P)&\ge \frac{1}{3}\left( \left( {\begin{array}{c}k\\ 2\end{array}}\right) (q+1) + k\cdot x\right) \cdot |P| \\ {}&\ge \frac{k}{6}\left( (k-1)(q+1) + x\right) \cdot |P| \\ {}&= \frac{k}{6}\left( |P|-q-2 \right) \cdot |P| \\ {}&= \frac{1}{6} \cdot \frac{|P|-x-1}{q+1} \cdot \left( |P|-q-2 \right) \cdot |P| \\ {}&\ge \frac{1}{6} \cdot \frac{|P|-q-2}{q+1} \cdot \left( |P|-q-2 \right) \cdot |P| \\ {}&\ge \frac{1}{6} \cdot \frac{|P|}{4q} \cdot \left( \frac{3|P|}{8} \right) \cdot |P| = \frac{|P|^3}{64q}. \end{aligned}$$

The final inequality uses the assumption that \(q \ge 8\). \(\square \)

We remark here that, if we take a closer inspection of the last step in the previous proof, we can carry out a small case analysis to check that the condition \(q \ge 8\) is not really needed. However, since we will need q to be sufficiently large in several other steps of our forthcoming arguments, we omit the details of this calculation. Moreover, the constant \(\frac{1}{64}\) in Corollary 2.4 can be improved further, at the expense of some slightly more involved calculations that we prefer to skip.

We now turn towards the proof of Lemma 2.2. The proof uses Karamata’s inequality. For two finite non-increasing sequences of real numbers \(x_1,\dots x_n\) and \(y_1,\dots y_n\), we say that \(x_1,\dots ,x_n\) majorizes \(y_1,\dots , y_n\) if

$$\begin{aligned} x_1+\dots + x_i \ge y_1+ \dots + y_i, \,\,\,\,\,\,\, \forall 1 \le i \le n \end{aligned}$$
(7)

and

$$\begin{aligned} x_1+\dots +x_n = y_1+ \dots +y_n. \end{aligned}$$
(8)

Lemma 2.5

(Karamata’s Inequalty) Let \(f:I \rightarrow \mathbb R\) be a convex function defined on an interval I. Suppose that \((x_i)_{i =1,\dots ,n}\) and \((y_i)_{i = 1, \dots n}\) are non-increasing sequences in I such that \((x_i)_{i =1,\dots ,n}\) majorizes \((y_i)_{i =1,\dots ,n}\). Then

$$\begin{aligned} \sum _{i=1}^{n}f(x_i) \ge \sum _{i=1}^{n}f(y_i). \end{aligned}$$

Proof of Lemma 2.2

For any point \(v\) in \({\mathbb {F}}_q^2\); there are exactly \((q+1)\) lines passing through it. Let the set of lines passing through \(v\) be \(\mathcal {L}(v)\). We label the lines in \({\mathcal {L}}(v)\) according the number of elements of P they contain, in non-increasing order. That is, we write

$$\begin{aligned} \mathcal {L}(v) = \{\ell _1^{(v)}, \dots , \ell _{q+1}^{(v)} \}, \end{aligned}$$

such that

$$\begin{aligned} i < j \Rightarrow |\ell _i^{(v)} \cap P| \ge |\ell _j^{(v)} \cap P|. \end{aligned}$$

We assign weights to points in \(P\) in a way such that the sum of weight of all the points in \(P\) will be T(P). For a point \(v\) in \(P\) assign a weight \(W(v)\) to it as follows:

$$\begin{aligned} W(v) := \frac{1}{3} \sum _{j=1}^{q+1} \left( {\begin{array}{c}w_j^{(v)}\\ 2\end{array}}\right) ; \end{aligned}$$
(9)

where \(w_j^{(v)} = |\ell _j^{(v)} \cap P|- 1\) for all \(j \in [q+1]\). Note that W(v) counts one third of the number of collinear triples in P which contain v. Therefore, assigning weights in this fashion satisfies the required property; that is,

$$\begin{aligned} T(P) = \sum _{v\in P} W(v). \end{aligned}$$
(10)

Also observe that for any \(v\in P\),

$$\begin{aligned} \sum _{j=1}^{q+1} w_{j}^{(v)} = |P|-1= k(q+1) + x. \end{aligned}$$
(11)

We will state and prove the following claim in order to finish the proof.

Claim 2.6

For any point \(v\) in \(P\),

$$\begin{aligned} W(v) \ge \frac{1}{3}\left( \left( {\begin{array}{c}k\\ 2\end{array}}\right) (q+1) + k\cdot x\right) . \end{aligned}$$

Proof of Claim

We observed in (11) that the sum \(\sum _{j=1}^{q+1} w_{j}^{(v)}\) is fixed, for all \(v \in P\). It’s a natural intuition from (9) that the quantity \(W(v)\) reaches its minimum when the \(w_{j}^{(v)}\) terms are almost equal, in other words \(|w_{j}^{(v)}-w_{j'}^{(v)}| \le 1\) for all \(j,j'\in [q+1]\). We will prove that this intuition is indeed true via an application of Karamata’s inequality.

The function \(f(y) = \left( {\begin{array}{c}y\\ 2\end{array}}\right) \) is convex, and the non-increasing sequence consisting of \(x\) occurrences of \((k+1)\) and \(q-x+1\) occurrences of \(k\) is majorized by every other non-increasing sequence of integers of length \(q+1\) and total sum \(k(q+1)+x\). It therefore follows from Karamata’s Inequality that

$$\begin{aligned} W(v)= & {} \frac{1}{3} \sum _{j=1}^{q+1} \left( {\begin{array}{c}w_{j}^{(v)}\\ 2\end{array}}\right) \ge \frac{1}{3} \left( \sum _{j=1}^{x} \left( {\begin{array}{c}k+1\\ 2\end{array}}\right) + \sum _{j=x+1}^{q+1} \left( {\begin{array}{c}k\\ 2\end{array}}\right) \right) \\= & {} \frac{1}{3}\left( \left( {\begin{array}{c}k\\ 2\end{array}}\right) (q+1) + k\cdot x\right) . \end{aligned}$$

This finishes the proof of the Claim 2.6. \(\square \)

Alternatively, instead of using Karamata’s inequality Claim 2.6 can also be proved by using the fact that

$$\begin{aligned} \left( {\begin{array}{c}y\\ 2\end{array}}\right) +\left( {\begin{array}{c}z\\ 2\end{array}}\right) \ge \left( {\begin{array}{c}y+i\\ 2\end{array}}\right) +\left( {\begin{array}{c}z-i\\ 2\end{array}}\right) \end{aligned}$$

for all \(z \ge z-i \ge y+i \ge y \ge 0\).

Now applying Claim 2.6 in (10) we conclude that

$$\begin{aligned} T(P)= \sum _{v\in P} W(v) \ge \sum _{v\in P}\frac{1}{3}\left( \left( {\begin{array}{c}k\\ 2\end{array}}\right) (q+1) + k\cdot x\right) = \frac{1}{3}\left( \left( {\begin{array}{c}k\\ 2\end{array}}\right) (q+1) + k\cdot x\right) \cdot |P|. \end{aligned}$$

Hence Lemma 2.2 is proved. \(\square \)

3 Container Lemmas for Arcs

3.1 Basic Properties of the Graph Encoding Triples

Define a 3-uniform hypergraph \({\mathcal {H}}\) with vertices corresponding to points in \({\mathbb {F}}_q^2\), with three points forming a hyperedge if they are collinear. Note that the number of edges in this graph is

$$\begin{aligned} q(q+1) \left( {\begin{array}{c}q\\ 3\end{array}}\right) <q^5. \end{aligned}$$

In this section, we will make repeated applications of the container theorem for this graph and its induced subgraphs until we obtain a family of containers for arcs in \(\mathbb F_q^2\) (i.e. a family \({\mathcal {C}}\) of subsets of \(\mathbb F_q^2\) with the property that, for any arc \(P \subset \mathbb F_q^2\), there exists \(C \in {\mathcal {C}}\) such that \(P \subset C\)) with the properties we need. Before starting this iterative process, we collect a few inequalities that will be used repeatedly in the proofs of the forthcoming three lemmas.

A reformulation of Corollary 2.4 states that, for any \(A \subset \mathbb F_q^2\),

$$\begin{aligned} |A| \le \max \{2q, 4q^{1/3} T(A)^{1/3} \}. \end{aligned}$$
(12)

It therefore follows from the definition (4) that

$$\begin{aligned} T(A) \ge q^2 \Rightarrow d( {\mathcal {H}}[A]) \ge \frac{3T(A)^{2/3}}{4q^{1/3}} . \end{aligned}$$
(13)

We will need to bound the quantity \(\Delta _2({\mathcal {H}}[A])\) to apply Theorem 2.1. A first observation is that, for any \(A \subset \mathbb F_q^2\),

$$\begin{aligned} \Delta _2({\mathcal {H}}[A]) \le q-2. \end{aligned}$$
(14)

Indeed, given a pair of points in the plane, the number of points in \(\mathbb F_q^2\) which are collinear with the given pair is \(q-2\), and (14) follows.

A better bound for \(\Delta _2({\mathcal {H}}[A])\) is available when T(A) is smaller. We have

$$\begin{aligned} \Delta _2({\mathcal {H}}[A]) \le 2(T(A))^{1/3}. \end{aligned}$$
(15)

Indeed,

$$\begin{aligned} \Delta _2({\mathcal {H}}[A])= \max _{\ell \in \{ \text {all lines in } \mathbb F_q^2 \}} | \ell \cap A| -2:=M-2. \end{aligned}$$

A line containing M elements of P gives rise to \(\left( {\begin{array}{c}M\\ 3\end{array}}\right) \) collinear triples, and so

$$\begin{aligned} T(A) \ge \left( {\begin{array}{c}M\\ 3\end{array}}\right) \ge \frac{(\Delta _2( {\mathcal {H}}[A]))^3}{6}. \end{aligned}$$

A rearrangement of this inequality gives (15).

3.2 A First Container Lemma for Arcs

The argument for our first container lemma largely follows that of [12, Lem. 2]. The only difference in what follows is that we keep track of the number of collinear triples determined by the containers, rather than their size.

Lemma 3.1

Suppose that q is a sufficiently large prime power. Then there exists a family \({\mathcal {C}}^1\) of subsets of \(\mathbb F_q^2\) such that

  • \(|{\mathcal {C}}^1| \le 2^{Kq^{2/3}(\log q)^2}\),

  • For all \(C \in {\mathcal {C}}^1\), \(T(C) \le q^{3}\),

  • For every arc \(P \subset \mathbb F_q^2\), there exists \(C \in {\mathcal {C}}^1\) such that \(P \subset C\).

In the first point above, K is an absolute constant.

Proof

In this proof, we will iteratively apply Theorem 2.1 to subsets of \(\mathbb F_q^2\). The structure of the argument is similar to that of the proof of the main result in [5]. We begin by applying it to the graph \({\mathcal {H}}\) encoding collinear triples, which we defined at the beginning of this section. Note that independent sets in this hypergraph are the same thing as arcs in \(\mathbb F_q^2\). As a result, we obtain a set \({\mathcal {C}}_1\) of containers. We iterate by considering each \(A \in {\mathcal {C}}_1\). If A contains too many collinear triples, then we apply Theorem 2.1 to the graph \({\mathcal {H}}[A]\) to get a family of containers \({\mathcal {C}}_A\). If the number of collinear triples in A is sufficiently small then we put this A into a final set \({\mathcal {C}}\) of containers (or to put it another way, we write \({\mathcal {C}}_A=\{A\}\)).

Repeating this for all \(A \in {\mathcal {C}}_1\) we obtain a new set of containers

$$\begin{aligned} {\mathcal {C}}_2 = \bigcup _{A \in {\mathcal {C}}_1} {\mathcal {C}}_A. \end{aligned}$$

Note that \({\mathcal {C}}_2\) is a container set for \({\mathcal {H}}\). Indeed, suppose that X is an independent set in \({\mathcal {H}}\). Then there is some \(A \in {\mathcal {C}}_1\) such that \(X \subset A\). Also, X is an independent set in the hypergraph \({\mathcal {H}}[A]\), which implies that \(X \subset A'\) for some \(A' \in {\mathcal {C}}_A \subset {\mathcal {C}}_2\).

We then repeat this process, defining

$$\begin{aligned} {\mathcal {C}}_i = \bigcup _{A \in {\mathcal {C}}_{i-1}} {\mathcal {C}}_A. \end{aligned}$$

By choosing the values of \(\tau \) and \(\epsilon \) appropriately, we can ensure that after relatively few steps we have \(T(A) \le q^3\) for all of the sets \(A \in {\mathcal {C}}_m\). We then declare \(\mathcal C^1= {\mathcal {C}}_m\). It turns out that, because of m being reasonably small, \(|{\mathcal {C}}^1|\) is also fairly small.

Fig. 1
figure 1

An illustration of the beginning of the construction of the set of containers \({\mathcal {C}}^1\), which can be viewed as a tree. For each \(A_i \in {\mathcal {C}}_{\mathbb F_q^2}\), we check the size of \(T(A_i)\), and if it is still too large we apply the container theorem again. If the set contains few enough collinear triples, we stop the process. This is what happens with the set \(A_2\) in the diagram. The final set of containers \({\mathcal {C}}^1\) consists of the leaves of the tree

Full size image

Now we give more precise details of how to run this argument. Let \(A \in {\mathcal {C}}_j\), with \(j \le m\), and write \(T(A)=q^{5-s}\). If \(s \ge 2\) then we add A to the final set of containers \({\mathcal {C}}^1\). Otherwise, we apply Theorem 2.1 to \({\mathcal {H}}[A]\). We choose the parameters \(\epsilon \) and \(\tau \) to be

$$\begin{aligned} \epsilon = \frac{1}{3}, \,\,\,\,\,\,\,\, \tau = \max \{ 3000q^{\frac{2s}{3}-2}, 100 q^{\frac{s}{3}-\frac{3}{2} }\}. \end{aligned}$$
(16)

In order for this application of Theorem 2.1 to be legitimate, we need to make some calculations involving d(H[A]) and \(\Delta _2(H[A])\), and check that the conditions of Theorem 2.1 are satisfied. The main challenge is to verify that (5) holds.

Since we have \(T(A) \ge q^3\), it follows from (13) that

$$\begin{aligned} d({\mathcal {H}}[A]) \ge \frac{3T(A)^{2/3}}{4q^{1/3}}=\frac{3}{4}q^{3-\frac{2s}{3}}. \end{aligned}$$
(17)

Applying (17) and (14) and then using the two components of the definition of \(\tau \), it follows that

$$\begin{aligned} \frac{\Delta _2({\mathcal {H}})}{d({\mathcal {H}}) \cdot \tau }+ \frac{1}{2d({\mathcal {H}}) \cdot \tau ^{2}}&\le \frac{4}{3q^{2-\frac{2s}{3}}\cdot \tau }+ \frac{1}{q^{3-\frac{2s}{3}} \cdot \tau ^{2}} \\ {}&\le \frac{4}{3 \cdot 3000} + \frac{1}{10000} \le \frac{\epsilon }{288}. \end{aligned}$$

We have therefore verified that (5) holds. Since \(s \le 2\), the condition that \( \tau <\frac{1}{2}\) is guaranteed to hold as long as we choose q to be sufficiently large. The condition \(\epsilon <1/2\) is immediate.

Theorem 2.1 therefore gives the bound

$$\begin{aligned} |{\mathcal {C}}_A| \le 2^{c |A|\tau \log q}, \end{aligned}$$
(18)

for some absolute constant c (which may be different from the absolute constant c in the statement of Theorem 2.1). Applying (12) and the fact that \(T(A)=q^{5-s} \ge q^3 \ge q^2\), we see that

$$\begin{aligned} |A| \le 4q^{2-s/3}. \end{aligned}$$
(19)

It then follows from (19), (18), the definition of \(\tau \), and the upper bound \(s \le 2\), that

$$\begin{aligned} |{\mathcal {C}}_A| \le 2^{c' q^{\frac{2}{3}} \log q}, \end{aligned}$$

for some absolute constant \(c'\).

Since each step of this process reduces the number of edges in the containers by a factor of 3, it follows that the process will terminate after at most \(2 \log _3 q\) steps. The final set \({\mathcal {C}}^1\) therefore contains at most

$$\begin{aligned} \left( 2^{c'q^{\frac{2}{3}} \log q} \right) ^{2 \log _3 q} = 2^{Kq^{\frac{2}{3}} (\log q)^2}, \end{aligned}$$

elements, for some absolute constant K. \(\square \)

3.3 A Second Container Lemma for Arcs

We use Lemma 3.1 as a basis for the following improved result, in which we reduce the number of edges further without paying too serious a price. The main quantitative cost of this reduction is an increase in the size of the final set of containers, and this increase depends on the new error-term parameter \(\gamma \). The proof of the following lemma again uses the supersaturation results established in Sect. 2, and this time we use both Corollaries 2.3 and 2.4.

Lemma 3.2

Suppose that q is a sufficiently large prime power. There exist positive absolute constants c and \(K'\) such that, for any \( \gamma \) in the range \(cq^{-1/2} \le \gamma \le 1\), following statement is true. There exists a family \({\mathcal {C}}^2\) of subsets of \(\mathbb F_q^2\) such that

  • \(|{\mathcal {C}}^2| \le 2^{K'\gamma ^{-\frac{2}{3}}q^{\frac{2}{3}}(\log q)^2}\),

  • For all \(C \in {\mathcal {C}}^2\), \(|C| \le (1+ \gamma )q\),

  • For every arc \(P \subset \mathbb F_q^2\), there exists \(C \in {\mathcal {C}}^2\) such that \(P \subset C\).

Proof

Let \({\mathcal {C}}^1\) be the set of containers given by Lemma 3.1, and fix \(C \in {\mathcal {C}}^1\). Similar to the proof of Lemma 3.1, we will iteratively apply Theorem 2.1 to obtain a family \({\mathcal {C}}^2\) of containers whose components contain fewer collinear triples.

Let A be a container obtained during this iterative process and write \(T(A)=\frac{\gamma }{6}q^{5-s}\). If \(s \ge 3\) then we add A to the final set of containers \({\mathcal {C}}^2\). Otherwise, we will apply Theorem 2.1 to \({\mathcal {H}}[A]\) to obtain a family \({\mathcal {C}}_A\) of containers for \({\mathcal {H}}[A]\).

The main difference between this proof and that of Lemma 3.1 is that we bound \(\Delta _2({\mathcal {H}}[A])\) using (15). Since \(T(A) \le q^3\), this bound is better than the bound (14) that we used in the proof of Lemma 3.1. In particular, we have

$$\begin{aligned} \Delta _2({\mathcal {H}}[A]) \le 2(T(A))^{1/3} \le 2 \gamma ^{1/3}q^{\frac{5}{3}-\frac{s}{3}}. \end{aligned}$$
(20)

Using (12), along with the fact that \(T(A) \le \gamma q^{5-s}\) and \(s \le 3\), yields

$$\begin{aligned} |A| \le \max \{2q, 4q^{1/3} T(A)^{1/3} \} \le \max \{2q, 4\gamma ^{\frac{1}{3}}q^{2- \frac{s}{3}} \} \le 4q^{2-\frac{s}{3}}. \end{aligned}$$
(21)

It therefore follows that

$$\begin{aligned} d( {\mathcal {H}}[A]) = \frac{3T(A)}{|A|} \ge \frac{\frac{\gamma }{2}q^{5-s}}{4q^{2-\frac{s}{3}}} = \frac{\gamma }{8} q^{3-\frac{2s}{3}}. \end{aligned}$$
(22)

We choose the parameters \(\epsilon \) and \(\tau \) to be

$$\begin{aligned} \epsilon = \frac{1}{3}, \,\,\,\,\,\,\,\, \tau =32000\gamma ^{-\frac{2}{3}}q^{\frac{s}{3}-\frac{4}{3}}. \end{aligned}$$
(23)

We again need to check that (5) holds. Indeed, by (20), (22), and the definition of \(\tau \),

$$\begin{aligned} \frac{\Delta _2({\mathcal {H}})}{d({\mathcal {H}}) \cdot \tau }+ \frac{1}{2d({\mathcal {H}}) \cdot \tau ^{2}} \le \frac{1}{2000}+ \frac{\gamma ^{1/3} q^{-\frac{1}{3}}}{1000000} \le \frac{1}{2000}+ \frac{1}{1000000} \le \frac{\epsilon }{288}. \end{aligned}$$

We have therefore verified that (5) holds. We also need to ensure that the condition \(\tau <1/2\) holds. This follows from the assumption that \(\gamma >cq^{-1/2}\), along with the fact that \(s \le 3\), provided that the absolute constant c is chosen to be sufficiently large. Indeed,

$$\begin{aligned} \tau =32000\gamma ^{-\frac{2}{3}}q^{\frac{s}{3}-\frac{4}{3}} \le \frac{32000}{c^{2/3}}q^{\frac{s}{3}-1} \le \frac{32000}{c^{2/3}} < \frac{1}{2}. \end{aligned}$$

Theorem 2.1 can therefore be legitimately applied, and it gives the bound

$$\begin{aligned} |{\mathcal {C}}_A| \le 2^{c'|A|\tau \log q}, \end{aligned}$$
(24)

for some absolute constant \(c'\). Applying (21) and the definition of \(\tau \), one obtains the bound

$$\begin{aligned} \tau |A| \le C \max \left\{ q^{\frac{s}{3}-\frac{1}{3}}\gamma ^{-\frac{2}{3}}, q^{\frac{2}{3}}\gamma ^{-1/3} \right\} \le C q^{\frac{2}{3}}\gamma ^{-\frac{2}{3}}, \end{aligned}$$

where C is an absolute constant. Combining this with (24) gives

$$\begin{aligned} |{\mathcal {C}}_A| \le 2^{c''\gamma ^{-\frac{2}{3}} q^{\frac{2}{3}} \log q}, \end{aligned}$$

with a new absolute constant \(c''\).

Since each step of this process reduces the number of collinear triples in the containers by a factor of 3, it follows that the process will comfortably terminate after at most \(2 \log _3 q\) steps. We also need to take into account that we already started with \(|{\mathcal {C}}^1|\) containers, and that we apply this process for each of the elements of \({\mathcal {C}}^1\). Therefore, the final set \({\mathcal {C}}^2\) contains at most

$$\begin{aligned} |{\mathcal {C}}^1|\left( 2^{c''\gamma ^{-\frac{2}{3}} q^{\frac{2}{3}} \log q}\right) ^{2 \log _3 q}{} & {} \le |{\mathcal {C}}^1| 2^{4c''\gamma ^{-\frac{2}{3}}q^{\frac{2}{3}} (\log q)^2} \le 2^{Kq^{\frac{2}{3}}(\log q)^2}2^{4c''\gamma ^{-\frac{2}{3}}q^{\frac{2}{3}} (\log q)^2} \\{} & {} \le 2^{K'\gamma ^{-\frac{2}{3}}q^{\frac{2}{3}} (\log q)^2} \end{aligned}$$

elements, where \(K'\) is an absolute constant.

It remains to check that \(|A| \le q(1+\gamma )\) for all \(A \in {\mathcal {C}}\). This follows immediately from Corollary 2.3. \(\square \)

4 Counting Arcs

We are now ready to prove the main results of the paper. We start with Theorem 1.1, which is restated below for convenience.

Theorem 4.1

There exists an absolute constant C such that, for all q sufficiently large,

$$\begin{aligned} |{\mathcal {A}}(q)| \le 2^{q+Cq^{\frac{4}{5}}(\log q)^2}. \end{aligned}$$

Proof

Let \({\mathcal {C}}^2\) denote the set of containers given by Lemma 3.2, applied with \(\gamma =q^{-\frac{1}{5}}\). All of the sets in \({\mathcal {A}}(q)\) are subsets of some \(C \in {\mathcal {C}}^2\). Since \(|C| \le (1+ \gamma )q = q + q^{\frac{4}{5}}\), it follows that

$$\begin{aligned} |{\mathcal {A}}(q)| \le |{\mathcal {C}}^2| 2^{q+q^{\frac{4}{5}}} \le 2^{K'q^{\frac{4}{5}}(\log q)^2}\cdot 2^{q+q^{\frac{4}{5}}} \le 2^{q+(K'+1)q^{\frac{4}{5}}(\log q)^2}. \end{aligned}$$

The result holds, with \(C=K'+1\). \(\square \)

Now we prove Theorem 1.2, which we will derive from the following more general statement.

Theorem 4.2

Let q be a sufficiently large prime power and let C is a sufficiently large absolute constant. Let \(k \in \mathbb N\) satisfy \(Cq^{2/3}(\log q)^2 \le k \le q\) and define \(\gamma =\frac{C^{1/2}q^{2/5}(\log q)^{6/5}}{k^{3/5}}\). Then

$$\begin{aligned} |{\mathcal {A}}(q,k)| \le \left( {\begin{array}{c}(1+2\gamma )q\\ k\end{array}}\right) . \end{aligned}$$

Proof

Apply Lemma 3.2 with this choice of \(\gamma \). The assumption that \(Cq^{2/3}(\log q)^2 \le k \le q\) is sufficient to ensure that the condition on \(\gamma \) in Lemma 3.2 is satisfied. All of the sets in \({\mathcal {A}}(q,k)\) are subsets of some \(C \in {\mathcal {C}}^2\). Since \(|C| \le (1+\gamma )q\), it follows that

$$\begin{aligned} |{\mathcal {A}}(q,k)|&\le |{\mathcal {C}}^2| \left( {\begin{array}{c} (1+\gamma )q\\ k\end{array}}\right) \\ {}&\le 2^{K'\gamma ^{-\frac{2}{3}}q^{\frac{2}{3}}( \log q)^2}\cdot \left( {\begin{array}{c} (1+\gamma )q\\ k\end{array}}\right) \\ {}&\le \left( {\begin{array}{c}(1+2\gamma )q\\ k\end{array}}\right) . \end{aligned}$$

The final inequality follows from the lower bound on k in the statement of the theorem and the assumption that C is sufficiently large (one may take \(C=K'\)). \(\square \)

Proof of Theorem 1.2

If we now apply Theorem 4.2 with the slightly stronger lower bound \(k \ge q^{2/3}(\log q)^3\), we see that \(\gamma = o(1)\), thus proving Theorem 1.2\(\square \)

Finally, we consider the size of the largest arc contained in a random point set, improving a result from [12]. Given \(P \subset \mathbb F_q^2\), let a(P) denote the size of the largest arc \(P'\) such that \(P' \subseteq P\). Let \(Q_p\) be a random subset of \(\mathbb {F}_{q}^{2}\) with the events \(x \in Q_p\) being independent with probability \(\mathbb {P}[x \in Q_p] = p\). We say that \(Q_p\) is a p-random set.

Theorem 4.3

Suppose that \(K'q^{-\frac{1}{3}}(\log q)^2 \le p\le 1\), where \(K'\) is the constant from Lemma 3.2, and let \(Q_p \subseteq \mathbb F_q^2\) be a p-random set. Then

$$\begin{aligned} \lim _{q \rightarrow \infty } \mathbb P [a(Q_p) \ge 11qp]= 0. \end{aligned}$$

By contrast, it was established in [12] that, with high probability, \(a(Q_p) =\Omega (qp)\). Combining this observation with Theorem 4.3, we see that a random set \(Q_p\) with p relatively large is very likely to have \(a(Q_p) = \Theta (qp)\).

Other ranges of p for this problem were also considered in [12], and a near-optimal bound for \(p<1/q\) was proven. The problem of determining the behaviour of \(a(Q_p)\) in the range \(q^{-1} \le p \le q^{-1/3}\) was very recently settled by Chen, Liu, Nie and Zeng [7].

Proof of Theorem 4.3

Fix p in the given range and let \(m:=\lfloor 11pq \rfloor -1 \). Apply Lemma 3.2 with \(\gamma =1\) to obtain a family of containers \({\mathcal {C}}^2\). For q sufficiently large, the probability that \(Q_p\) contains an arc of size m is at most

$$\begin{aligned} |{\mathcal {C}}^2| \left( {\begin{array}{c}2q\\ m\end{array}}\right) p^m. \end{aligned}$$

This is because an arc of size m must be contained in some \(C \in {\mathcal {C}}^2\), and each subset of size m belongs to the random subset \(Q_p\) with probability \(p^m\). Applying the bounds from Lemma 3.2, as well as the bound \(\left( {\begin{array}{c}a\\ b\end{array}}\right) \le \left( \frac{ea}{b} \right) ^b\), it follows that

$$\begin{aligned} \mathbb P [ a(Q_p) \ge 11pq] \le \mathbb P [ a(Q_p) \ge m]&\le |{\mathcal {C}}^2| \left( {\begin{array}{c}2q\\ m\end{array}}\right) p^m \\ {}&\le 2^{K'q^{2/3}(\log q)^2} \cdot \left( \frac{2eqp}{m} \right) ^m \\ {}&\le \left( \frac{4eqp}{m} \right) ^m \\ {}&\le (0.99)^m. \end{aligned}$$

The second from last of these inequalities is an application of the inequality \(K'q^{2/3}(\log q)^2 \le m\), which follows from the lower bound on p assumed in the statement of the theorem. It follows that

$$\begin{aligned} \lim _{q \rightarrow \infty } \mathbb P [ a(Q_p) \ge 11pq] =0. \end{aligned}$$

\(\square \)