# Two Upper Bounds for the Erdős–Szekeres Number with Conditions

## Abstract

We use a projective transformation method of Tóth and Valtr to show that a certain number of points in general position in the plane contain the vertex set of a convex \(n\)-gon if their convex hull is an \((n\!-\!1)\)-gon.

## Keywords

Erdős-Szekeres problem Discrete geometry Combinatorial convexity## 1

In 1933 the following question was asked, inspired by Esther Klein.

Let \(n\ge 3\) be an integer. Is there a finite number, of which the smallest is denoted by \(\text{ ES}(n)\) in the following, such that every set of \(\text{ ES}(n)\) points in general position in the plane contains the vertex set of a convex \(n\)-gon?

*Erdős–Szekeres conjecture*. The actual best upper bound has been given by Tóth and Valtr [8]. They proved that for all \(n\ge 5\)

In this paper we use the projective transformation method of Tóth and Valtr to show the following statements.

### **Theorem 1**

For \(n\ge 4\), every set of \(\genfrac(){0.0pt}{}{2n-5}{n-2}-2n+9\) points in general position in the plane with the property that there is no point inside a triangle determined by three consecutive vertices of its convex hull contains a convex \(n\)-gon.

### **Theorem 2**

For \(n\ge 4\), every set of \(\genfrac(){0.0pt}{}{2n-5}{n-2}-n-(\sum _{i=1}^{n-4}\min \{i,s\})+6\) points in general position in the plane with an \((n\!-\!1)\)-gon as its convex hull contains a convex \(n\)-gon where \(s\ge 0\) denotes the number of points inside a triangle determined by three consecutive vertices of its convex hull (Fig. 1).

### **Corollary 3**

For \(n\ge 4\), every set of \(\genfrac(){0.0pt}{}{2n-5}{n-2}-2n+10\) points in general position in the plane with an \((n\!-\!1)\)-gon as its convex hull contains a convex \(n\)-gon.

### **Definition 4**

A tuple \((p_1,p_2,\ldots ,p_n)\) of \(n\) points in the plane is called a *convex* \(n\) *-gon* if none of its points is contained in the convex hull of the others and the counterclockwise radial order around \(p_1\) starting outside the convex hull is \((p_2,p_3,\ldots ,p_n)\).

The above mentioned results all make use of special convex \(n\)-gons, the \(n\)-caps and \(n\)-cups.

### **Definition 5**

*-cap*if

*-cup*if

### **Lemma 6**

A key idea of Tóth and Valtr was to transform the considered set of points into a special one without affecting the convexity relations between the points.

### **Definition 7**

*special point set*is a finite set \(M\) of points in the plane together with one of its elements \(a_1\), convex hull \((a_1,a_2,\ldots ,a_k)\) and \(|M|\ge 3\) such that

- (1)
no three of its points are on a common line,

- (2)
the point \(a_1\) has strictly the biggest \(y\)-coordinate,

- (3)
only \(a_1\) and \(a_3\) have the same \(x\)-coordinate which is zero,

- (4)
no line determined by any two points of \(M\setminus \{a_1,a_3\}\) crosses the ray above \(a_1\) on the line \(a_1a_3\).

*upmost point*of the special point set and the set \(M\setminus \{a_1\}\) is called the associated

*reduced special point set*(Fig. 3).

The crucial advantage of considering a special point set is property (4). It guarantees that every \((n\!-\!1)\)-cup of the associated reduced special point set can be extended by the upmost point \(a_1\) to a convex \(n\)-gon. Properties (2) and (3) are only added for convenience and were not used by Tóth and Valtr in their proof.

### **Lemma 8**

Let \(M\) be a finite set of points in the plane with no three points on a common line, let \(|M|\ge 3\) and let \(a_1\) be a point of its convex hull. Then there exists a bijection \(\varphi :M\rightarrow M^{\prime }\) where \((M^{\prime },\varphi (a_1))\) is a special point set and \(\varphi \) respects the convexity relations, i. e. a point \(p\) is in the convex hull of the points \(p_1,p_2,\ldots ,p_n\) if and only if \(\varphi (p)\) is in the convex hull of \(\varphi (p_1),\varphi (p_2),\ldots ,\varphi (p_n)\).

### *Proof*

Since scaling and rotation of a point set do not change the convexity relations between the points, one may assume that \(M\) has properties (2) and (3). Now one proceeds as in [7]. Move the set without changing the \(x\)-coordinates such that \(a_1\) has a strictly negative \(y\)-coordinate and no line determined by any two points of \(M\setminus \{a_1,a_3\}\) crosses the segment between \(a_1\) and \((0,0)\). The projective transformation mapping the \(x\)-axis to the line at infinity along the \(y\)-axis gives the desired special point set \((M^{\prime },\varphi (a_1))\). The transformation on the lower halfplane is explicitly given by \(\varphi (x,y)=(x/y,1/y)\). Hence, property (2) still holds for the transformed set. Moreover, \(\varphi (a_1)\) and \(\varphi (a_3)\) have the same \(x\)-coordinate and by slightly moving the point set \(M\) upwards before the transformation, property (3) holds and \((M^{\prime },\varphi (a_1))\) is a special point set fulfilling the convexity condition. \(\square \)

### **Definition 9**

Let \((M,a_1)\) be a special point set with convex hull \((a_1,a_2,\ldots ,a_k)\). A tuple \((p,p_1,\ldots ,p_{a-1})\) is called an \(a\) *-ecap* if it is either an \(a\)-cap or if it is a convex \(a\)-gon, \(p=a_3\) and \((p_1,\ldots ,p_{a-1})\) is an \((a\!-\!1)\)-cap.

### *Proof of Theorem 1*

Let \(M\) be a set of \(\genfrac(){0.0pt}{}{2n-5}{n-2}-2n+9\) points in general position in the plane with the property that there is no point inside the triangle determined by the consecutive vertices \(a_1,a_2,a_3\) of its convex hull \((a_1,a_2,\ldots ,a_k)\). Using Lemma 8, we may assume that \((M,a_1)\) is a special point set with only \(a_2\) having a negative \(x\)-coordinate.

### *Proof of Theorem 2*

Let \(M\) be a set of \(\genfrac(){0.0pt}{}{2n-5}{n-2}-n-(\sum _{i=1}^{n-4}\min \{i,s\})+6\) points in general position in the plane with an \((n\!-\!1)\)-gon as its convex hull \((a_1,a_2,\ldots ,a_{n-1})\) and with a set \(S\) of \(s\) points inside the triangle determined by the points \(a_1,a_2,a_3\) (see Fig. 1). Using Lemma 8 we may assume that \((M,a_1)\) is a special point set.

## Notes

### Acknowledgments

The author would like to thank the two anonymous referees for helpful remarks.

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