# Many Collinear \(k\)-Tuples with no \(k+1\) Collinear Points

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## Abstract

For every \(k>3\), we give a construction of planar point sets with many collinear \(k\)-tuples and no collinear \((k+1)\)-tuples. We show that there are \(n_0=n_0(k)\) and \(c=c(k)\) such that if \(n\ge n_0\), then there exists a set of \(n\) points in the plane that does not contain \(k+1\) points on a line, but it contains at least \(n^{2-({c}/{\sqrt{\log n}})}\) collinear \(k\)-tuples of points. Thus, we significantly improve the previously best known lower bound for the largest number of collinear \(k\)-tuples in such a set, and get reasonably close to the trivial upper bound \(O(n^2)\).

## Keywords

Collinear Planar Point sets Construction## 1 Introduction

In the early 60s Paul Erdős asked the following question about point-line incidences in the plane: *Is it possible that a planar point set contains many collinear four-tuples, but it contains no five points on a line?* There are constructions for \(n\)-element point sets with \(n^2/6-O(n)\) collinear triples with no four on a line (see [4] or [11]). However, no similar construction is known for larger tuples.

### 1.1 Earlier Results and Our Result

This problem was among Erdős’ favourite geometric problems, he frequently talked about it and listed it among the open problems in geometry, see [6, 7, 8, 9, 10]. It is not just a simple puzzle which might be hard to solve, it is related to some deep and difficult problems in other fields. It seems that the key to tackle this question would be to understand the group structure behind point sets with many collinear triples. A recent result of Green and Tao—proving the Motzkin–Dirac conjecture [12]—might be an important development in this direction.

In the present paper, our goal is to give a construction showing that Erdős conjecture, if true, is sharp; for \(k>3\), one cannot replace the exponent 2 by \(2-c\), for any \(c>0\).

The first result was due to Kárteszi [15] who proved that \(t_k(n)\ge c_kn\log {n}\) for all \(k>3\). In 1976 Grünbaum [13] showed that \(t_k(n)\ge c_kn^{1+1/(k-2)}\). For some 30 years this was the best bound when Ismailescu [14], Brass et al. [2], and Elkies [5] consecutively improved Grünbaum’s bound for \(k\ge 5\). However, similarly to Grünbaum’s bound, the exponent was going to 1 as \(k\) went to infinity.

In what follows we are going to give a construction that substantially improves the lower bound. Namely, we will show the following.

### **Theorem 1**

For any \(k\ge 4\) integer, there is a positive integer \(n_0\) such that for \(n>n_0\) we have \(t_k(n)>n^{2-({c}/{\sqrt{\log n}})}\), where \(c=2\log (4k+9)\).

We note that each of the collinear \(k\)-tuples that we count in our construction has an additional property that the points form a \(k\)-term arithmetic progression, as the distance between every two consecutive points is the same in every coordinate.

### 1.2 Preliminaries

For \(r>0\) and a positive integer \(d\) let \(B_d(r)\) denote the closed ball in \(\mathbb R ^d\) of radius \(r\) centred at the origin, and \(S_d(r)\) denotes the sphere in \(\mathbb R ^d\) of radius \(r\) centred at the origin.

For any positive integer \(d\) the \(d\)-dimensional unit hypercube (and its translates) are denoted by \(H_d\). If the center of the cube is a point \(\mathbf{x}\in \mathbb R ^d\) then it has the vertex set \(\mathbf{x}+\{-1/2,1/2\}^d\) and we denote it by \(H_d(\mathbf{x})\).

For a set \(S\subseteq \mathbb R ^d\), let \(N(S)\) denote the number of points from the integer lattice \(\mathbb Z ^d\) that belong to \(S\), i.e., \(N(S):=|\mathbb Z ^d \cap S|\).

Throughout the paper the \(\log \) notation stands for the base 2 logarithm.

## 2 A Lower Bound on \(t_k(n)\)

We will prove bounds for even and odd value of \(k\) separately, as the odd case needs a bit more attention. Our proof is elementary, we use the fact that the volume of a large sphere approximates well the number of lattice points inside the ball. There are more advanced techniques to count lattice points on the surface of a sphere, however we see no way to improve our bound significantly by applying them.

In our construction, we rely on the fact that a point set in a large dimensional space that satisfies our collinearity conditions can be converted to a planar point set by simply projecting it to a plane along a suitably chosen vector, with all the collinearities preserved. That enables us to perform most of the construction in a space of large dimension, exploiting the properties of such a space.

### 2.1 Proof for Even \(k\)

### **Lemma 1**

### *Proof*

For every lattice point \(p\in B_d(r) \cap \mathbb Z ^d\) we look at the unit cube \(H_d(p)\) with center \(p\). These cubes all have disjoint interiors and each of them has diameter \(\sqrt{d}\). Moreover, their union \(\bigcup _{p\in B_d(r) \cap \mathbb Z ^d} H_d(p)\) is included in \(B_d(r+\sqrt{d}/2)\), it contains \(B_d(r-\sqrt{d}/2)\) and its volume is equal to \(N(B_d(r))\), which readily implies the statement of the lemma.\(\square \)

We will also use a bound on the number of points on a sphere.

### **Lemma 2**

### *Proof*

### *Proof*

(*of Theorem *1* for even* \(k\)) We will give a construction of a point set \(P\) containing no \(k+1\) collinear points, with a high value of \(t_k(P)\).

To obtain a point set in two dimensions, we project the \(d\) dimensional point set to a two dimensional plane in \(\mathbb R ^d\). The vector \(v\) along which we project should be chosen generically, so that every two points from our point set are mapped to different points, and every three points that are not collinear are mapped to points that are still not collinear.\(\square \)

### 2.2 Proof for Odd \(k\)

### *Proof*

(*of Theorem *1* for odd* \(k\)) We will give a construction of a point set \(P\) containing no \(k+1\) collinear points, with a high value of \(t_k(P)\).

On the other hand, every pair of points \(p_1,q_1\in \mathbb Z ^d \cap S_d(r)\) with different first coordinate, with \(d(p_1,q_1)=2\ell \), and with the middle point that belongs to \(\alpha _{x_0} \cap M\), defines one line that contains \(k\) points from \(P\). Note that such line cannot belong to \(\alpha _{x_0}\), as the first coordinates of \(p_1\) and \(q_1\) cannot be \(x_0\) simultaneously.

To obtain a point set in two dimensions, we project the \(d\) dimensional point set to a two dimensional plane in \(\mathbb R ^d\), similarly as in the even case.\(\square \)

## Notes

### Acknowledgments

The results presented in this paper are obtained during the authors’ participation in 8th Gremo Workshop on Open Problems—GWOP 2010. We thank the organizers for inviting us to the workshop and providing us with a gratifying working environment. Also, we are grateful for the inspiring conversations with the members of the group of Emo Welzl. József Solymosi is supported by NSERC, ERC-AdG. 321104, and OTKA NK 104183 grants. Miloš Stojaković is partly supported by Ministry of Science and Technological Development, Republic of Serbia, and Provincial Secretariat for Science, Province of Vojvodina.

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