Discrete & Computational Geometry

, Volume 49, Issue 2, pp 183–188

Two Upper Bounds for the Erdős–Szekeres Number with Conditions

• Florian Strunk
Article

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 and Szekeres answered this question positively in their joint paper [2]. More precisely, they proved in [2, 3] that for all integers $$n\ge 3$$ one has
\begin{aligned} 2^{n-2}+1\le \text{ ES}(n)\le \genfrac(){0.0pt}{}{2n-4}{n-2}+1. \end{aligned}
The lower bound is sharp for $$n\in \{3,4,5,6\}$$ where the case $$n=6$$ was treated recently in [6] with the help of computers. Erdős and Szekeres conjectured that the lower bound is tight in general which is known as the Erdős–Szekeres conjecture. The actual best upper bound has been given by Tóth and Valtr [8]. They proved that for all $$n\ge 5$$
\begin{aligned} \text{ ES}(n)\le \genfrac(){0.0pt}{}{2n-5}{n-2}+1. \end{aligned}
This is approximately an improvement by the factor two in relation to the upper bound of Erdős and Szekeres. The result of Tóth and Valtr combines two previous methods. One method has been applied by Chung and Graham [1] to lower the upper bound of Erdős and Szekeres by one. The second method was used by Tóth and Valtr themselves in [7] where they were able to show that $$\text{ ES}(n)\le \genfrac(){0.0pt}{}{2n-5}{n-2}+2$$ by using a projective transformation method. In the same year Kleitman and Pachter [4] proved that $$\text{ ES}(n)\le \genfrac(){0.0pt}{}{2n-4}{n-2}-2n+7$$ for all $$n\ge 4$$. Unfortunately their method seems to resist a combination with the projective transformation idea of Tóth and Valtr in general. A survey of results concerning the Erdős–Szekeres conjecture can be found in [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 $$(p_2,p_3,\ldots ,p_n)$$.

By abuse of notation, we identify a convex $$n$$-gon $$(p_1,p_2,\ldots ,p_n)$$ with its cyclic permutations and the convex hull of a finite set of points in the plane with the $$n$$-gon given by its vertices.

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

Definition 5

Let $$p_1=(x_1,y_1),p_2=(x_2,y_2),\ldots , p_n=(x_n,y_n)$$ be points in the plane with $$x_1<x_2<\cdots < x_n$$. The tuple $$(p_1,p_2,\ldots ,p_n)$$ is called an $$n$$ -cap if
\begin{aligned} \frac{y_2-y_1}{x_2-x_1}>\frac{y_3-y_2}{x_3-x_2}>\cdots >\frac{y_n-y_{n-1}}{x_n-x_{n-1}}. \end{aligned}
Analogously $$(p_1,p_2,\ldots ,p_n)$$ is called an $$n$$ -cup if
\begin{aligned} \frac{y_2-y_1}{x_2-x_1}<\frac{y_3-y_2}{x_3-x_2}<\cdots <\frac{y_n-y_{n-1}}{x_n-x_{n-1}}. \end{aligned}
For $$a,b\ge 2$$ let $$f(a,b)$$ denote the least number such that every set of $$f(a,b)$$ points in general position in the plane contains an $$a$$-cap or a $$b$$-cup (Fig. 2). One may always fix a coordinate system such that no $$x$$-coordinates of the relevant points are equal. Since every $$n$$-cap and every $$n$$-cup is in particular a convex $$n$$-gon, one has $$\text{ ES}(n)\le f(n,n)$$. An advantage of considering $$n$$-caps and $$n$$-cups instead of general convex $$n$$-gons is the following: Suppose the rightmost point of an $$a$$-cap is simultaneously the leftmost point of a $$b$$-cup, then one may extend either the cap to an $$(a\!+\!1)$$-cap or the cup to a $$(b\!+\!1)$$-cup. This observation led to the following lemma of Erdős and Szekeres providing their upper bound to the number $$\text{ ES}(n)$$.

Lemma 6

([2, 3]) For $$a,b\ge 2$$ one has
\begin{aligned} f(a,b)=\genfrac(){0.0pt}{}{a+b-4}{a-2}+1. \end{aligned}

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

A 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. (1)

no three of its points are on a common line,

2. (2)

the point $$a_1$$ has strictly the biggest $$y$$-coordinate,

3. (3)

only $$a_1$$ and $$a_3$$ have the same $$x$$-coordinate which is zero,

4. (4)

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

The point $$a_1$$ is called the 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.

For $$a,b \ge 3$$ let $$g(a,b)$$ be the least number such that every reduced special point set with $$g(a,b)$$ points, associated to a special point set with upmost point $$a_1$$ and only $$a_2$$ having a negative $$x$$-coordinate, contains an $$a$$-ecap or a $$b$$-cup (without $$a_1$$). The statement of the Theorem follows from the relation
\begin{aligned} g(a,b)\le \genfrac(){0.0pt}{}{a+b-4}{a-2}-(a-3)-(b-3)+1 \end{aligned}
for the special case $$a=n$$ and $$b=(n\!-\!1)$$ as every $$n$$-ecap is a convex $$n$$-gon and every $$(n\!-\!1)$$-cup (without $$a_1$$) can be extended to a convex $$n$$-gon by $$a_1$$. Since $$g(3,b)= 3$$ for any $$b\ge 3$$, this is implied by an induction over $$a\ge 3$$ by the relation
\begin{aligned} g(a,b)\le g(a-1,b)+f(a,b-1)-2, \end{aligned}
which we are now going to prove.
Let $$a\ge 4$$, $$b\ge 3$$, and $$M^{\prime }$$ be a reduced special point set with cardinality at least $$g(a\!-\!1,b)+f(a,b\!-\!1)-2$$ associated to a special point set with upmost point $$a_1$$ and only $$a_2$$ having a negative $$x$$-coordinate. Let $$R$$ be the set of rightmost points of $$(a\!-\!1)$$-ecaps in $$M^{\prime }$$. Since $$a\ge 4$$, the point $$a_3$$ does not belong to $$R$$ and the set $$M^{\prime }\setminus R$$ is a reduced special point set with only $$a_2$$ having a negative $$x$$-coordinate. If $$M^{\prime }\setminus R$$ contains a $$b$$-cup we are done. Moreover, $$M^{\prime }\setminus R$$ contains no $$(a\!-1\!)$$-ecap. Therefore we have
\begin{aligned} |M^{\prime }\setminus R|\le g(a-1,b)-1 \end{aligned}
and hence $$|R|\ge f(a,b\!-\!1)-1$$. Set $$R^{\prime }=R\cup \{a_3\}$$. If $$R^{\prime }$$ contains an $$a$$-cap we are done. Hence, assume that $$R^{\prime }$$ contains a $$(b\!-\!1)$$-cup. If the leftmost point of this $$(b\!-\!1)$$-cup is in $$R$$, it is the rightmost point of an $$(a\!-\!1)$$-ecap. By the usual argument we could then either extend the ecap or the cup and we are done. If the leftmost point of this cup is $$a_3$$ then it can be extended by $$a_2$$. $$\square$$

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.

For $$a,b \ge 3$$ let $$h(a,b)$$ be the least number such that every reduced special point set with $$h(a,b)$$ points, associated to a special point set with upmost point $$a_1$$ and with the property that there are at least $$\min \{b-3,s\}$$ points inside the triangle determined by $$a_1,a_2,a_3$$, contains an $$a$$-ecap or a $$b$$-cup (without $$a_1$$). The statement follows from the relation
\begin{aligned} h(a,b)\le \genfrac(){0.0pt}{}{a+b-4}{a-2}-(b-3)-\left(\sum _{i=1}^{b-3}\min \{i,s\}\right)+1 \end{aligned}
for the special case $$a=n$$ and $$b=(n\!-\!1)$$. Since $$h(a,3)= a$$ for any $$a\ge 3$$, this is implied by an induction over $$b\ge 3$$ by the relation
\begin{aligned} h(a,b)\le h(a,b-1)+f(a-1,b)-2-\min \{b-3,s\}. \end{aligned}
Let $$a\ge 3$$, $$b\ge 4$$, and $$M^{\prime }$$ be a reduced special point set with cardinality at least $$h(a,b\!-\!1)+f(a\!-\!1,b)-2-\min \{b-3,s\}$$, associated to a special point set with upmost point $$a_1$$, convex hull $$(a_1,\ldots ,a_{k})$$ where $$k\ge b$$ and with the property that there are at least $$\min \{b-3,s\}$$ points inside the triangle determined by $$a_1,a_2,a_3$$. If $$k>b$$ we are done. Hence we may assume that $$M$$ has convex hull $$(a_1,\ldots ,a_{b})$$. Let $$R$$ be the set of rightmost points of $$(b\!-\!1)$$-cups in $$M^{\prime }$$. The points $$a_1,\ldots ,a_{b-1}$$ do not belong to $$R$$ because otherwise we could extend the associated $$(b\!-\!1)$$-cup. Since $$b\ge 4$$ the set $$M^{\prime }\setminus R$$ is a reduced special point set associated to $$M\setminus R$$ which has convex hull $$(a_1,\ldots ,a_{b-1},a^{\prime }_{b}\ldots ,a^{\prime }_{l})$$ with $$l\ge (b\!-\!1)$$ and which has at least $$\min \{b-4,s\}$$ points inside the triangle determined by $$a_1,a_2,a_3$$ as it is guaranteed that at least the $$\min \{b-3,s\}$$ leftmost points of $$S$$ are not rightmost points of some $$(b\!-\!1)$$-cup in $$M^{\prime }$$. If $$M^{\prime }\setminus R$$ contains an $$a$$-ecap we are done. Since $$M^{\prime }\setminus R$$ contains no $$(b\!-\!1)$$-cup either, one has
\begin{aligned} |M^{\prime }\setminus R|\le h(a,b-1)-1 \end{aligned}
and therefore $$|R|\ge f(a\!-\!1,b)-1-\min \{b-3,s\}$$. Set $$R^{\prime }=R\cup \{a_2\}\cup S$$. Written as a disjoint union, this is $$R^{\prime }=R\cup \{a_2\}\cup (S\setminus (S\cap R))$$ and therefore $$|R^{\prime }|\ge f(a\!-\!1,b)$$. If $$R^{\prime }$$ contains a $$b$$-cup we are done. Hence assume that $$R^{\prime }$$ contains an $$(a\!-\!1)$$-cap. If the leftmost point of this $$(a\!-\!1)$$-cap is in $$R$$, it is the rightmost point of a $$(b\!-\!1)$$-cup (without $$a_1$$). By the usual argument we could then either extend the cap or the cup and we are done. If the leftmost point of this $$(a\!-\!1)$$-cap $$(p_1,\ldots ,p_{a-1})$$ is in $$\{a_2\}\cup S$$, then the line $$p_{a-2}p_{a-1}$$ either intersects the segment $$a_2a_3$$ or it does not. In the first case $$(p_{a-2},p_{a-1},a_3,\ldots ,a_b)$$ is a $$b$$-cup in $$M^{\prime }$$ (see Fig. 3). In the second case the tuple $$(a_3,p_1,\ldots ,p_{a-1})$$ is an $$a$$-ecap and we are done (see Fig. 4).$$\square$$

Notes

Acknowledgments

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

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