Note: Professor Pach's number: [045]; Also in: Facsimile edition: World Classics in Mathematics Series, vol. 28, China Science Press, Beijing, 2006. Mimeographed from 100 Research Problems in Discrete Geometry (with W. O. J. Moser). Reference DCG-BOOK-2008-001 URL: http://www.math.nyu.edu/~pach/research_problems.html Record created on 2008-11-18, modified on 2017-05-12

Research Problems in Discrete Geometry

We survey the theory of increasing and decreasing subsequences of permutations.
Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic
behavior of the expected value of the length is(w) of the longest increasing subsequence of a
permutation w of 1, 2, . . . , n was obtained by Vershik�Kerov and (almost) by Logan�Shepp.
The entire limiting distribution of is(w) was then determined by Baik, Deift, and Johansson.
These techniques can be applied to other classes of permutations, such as involutions, and are
related to the distribution of eigenvalues of elements of the classical groups. A number of
generalizations and variations of increasing/decreasing subsequences are discussed, including
the theory of pattern avoidance, unimodal and alternating subsequences, and crossings and
nestings of matchings and set partitions.

/pdf/increasing-and-decreasing-subsequences-and-their-variants-6ct19uxmz2.pdf

Increasing and decreasing subsequences and their variants

Erdős asked whether every sufficiently large set of points in general position in the plane contains six points that form a convex hexagon without any points from the set in its interior. Such a configuration is called an empty convex hexagon. In this paper, we answer the question in the affirmative. We show that every set that contains the vertex set of a convex 9-gon also contains an empty convex hexagon.

/pdf/empty-convex-hexagons-in-planar-point-sets-5c8aaagabq.pdf

Empty Convex Hexagons in Planar Point Sets

Let P be a finite set of points in general position in the plane Let C(P) be the convex hull of P and let CiP be the ith convex layer of P A minimal convex set S of P is a convex subset of P such that every convex set of P ∩ C(S) different from S has cardinality strictly less than |S| Our main theorem states that P contains an empty convex hexagon if C1P is minimal and C4P is not empty Combined with the Erdos-Szekeres theorem, this result implies that every set P with sufficiently many points contains an empty convex hexagon, giving an affirmative answer to a question posed by Erdos in 1977

/pdf/the-empty-hexagon-theorem-55v7x4yacn.pdf

The Empty Hexagon Theorem

We describe a computer proof of the 17-point version of a conjecture originally made by Klein-Szekeres in 1932 (now commonly known as the "Happy End Problem") that a planar configuration of 17 points, no 3 points collinear, always contains a convex 6-subset. The proof makes use of a combinatorial model of planar configurations, expressed in terms of signature functions satisfying certain simple necessary conditions. The proof is more general than the original conjecture as the signature functions examined represent a larger set of configurations than those which are realisable. Three independent implementations of the computer proof have been developed, establishing that the result is readily reproducible.

/pdf/computer-solution-to-the-17-point-erdos-szekeres-problem-1jkr36l749.pdf

Computer solution to the 17-point Erdős-Szekeres problem

In 1935 Erdős and Szekeres proved that for any integer n ≥ 3 there exists a smallest positive integer N(n) such that any set of at least N(n) points in general position in the plane contains n points that are the vertices of a convex n-gon. They also posed the problem to determine the value of N(n) and conjectured that N(n) = 2n−2 + 1 for all n ≥ 3. Despite the efforts of many mathematicians, the Erdős-Szekeres problem is still far from being solved. This paper surveys the known results and questions related to the Erdős-Szekeres problem in the plane and higher dimensions, as well as its generalizations for the cases of families of convex bodies and the abstract convexity setting.

/pdf/the-erdos-szekeres-problem-on-points-in-convex-position-a-5486m5zpf2.pdf

V. Soltan

Papers

The Erdos-Szekeres problem on points in convex position – a survey