Erdős–szekeres problem

Definition

The erdős–szekeres problem is a well-known question in combinatorial geometry that asks for the minimum number of points in general position in the plane, needed to guarantee a specific number of collinear points. This problem highlights the interplay between combinatorial structures and geometric configurations, leading to significant discussions on open problems and conjectures in Ramsey Theory.

5 Must Know Facts For Your Next Test

1. The problem was posed independently by mathematicians Paul Erdős and László Szekeres in 1935.
2. It can be expressed in terms of sequences and states that for any integer n, there exists a minimum number of points that guarantees either n points forming a convex polygon or n points being collinear.
3. The smallest number of points needed to guarantee a certain configuration is denoted as the Erdős–Szekeres number, typically represented as $ES(n)$.
4. The Erdős–Szekeres problem relates to finding monotonic subsequences within sequences of numbers, which is crucial in various areas of combinatorics.
5. The problem is an example of how geometric configurations can lead to deeper questions in combinatorial theory, with connections to various conjectures still being explored today.

Review Questions

• How does the erdős–szekeres problem illustrate the relationship between geometry and combinatorics?
  • The erdős–szekeres problem showcases the connection between geometry and combinatorics by asking how many points are needed to ensure specific geometric configurations, such as collinear points or vertices of convex polygons. This interplay leads to various combinatorial arguments about arrangements and sequences, illustrating how geometric questions can translate into combinatorial problems. The exploration of such problems encourages mathematicians to discover new relationships and patterns within these fields.
• Discuss the significance of the Erdős–Szekeres number and its implications for combinatorial geometry.
  • The Erdős–Szekeres number is significant because it provides a concrete threshold for determining how many points are necessary to ensure certain geometric arrangements, like convex polygons or collinear sets. It highlights the foundational concepts in combinatorial geometry and has implications for understanding larger structures in mathematics. By studying these numbers, researchers can better grasp how points interact in various configurations, leading to further exploration of related conjectures and open problems.
• Evaluate the impact of the erdős–szekeres problem on ongoing research in Ramsey Theory and related areas.
  • The erdős–szekeres problem has had a profound impact on ongoing research in Ramsey Theory by presenting a tangible question that bridges geometry and combinatorics. Its implications extend beyond just counting points; it influences various conjectures and helps mathematicians explore new avenues in structural properties of combinations. The problem encourages inquiry into broader themes, such as monotonic sequences and configuration spaces, thereby advancing our understanding of these complex mathematical landscapes.

Related terms

Collinear Points: Points that lie on the same straight line.

General Position: A configuration of points where no three points are collinear and no two points coincide.

Ramsey Theory: A branch of mathematics that studies conditions under which a certain structure must appear within a larger structure, often dealing with combinatorial problems.