There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The use of randomness is now an accepted tool in Theoretical Computer Science but not everyone is aware of the underpinnings of this methodology in Combinatorics - particularly, in what is now called the probabilistic Method as developed primarily by Paul Erdoős over the past half century. Here I will explore a particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets. A central point will be the evolution of these problems from the purely existential proofs of Erdős to the algorithmic aspects of much interest to this audience.

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The Probabilistic Method

An $(n,s)$ Davenport--Schinzel sequence, for positive integers $n$ and $s$, is a sequence composed of $n$ symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly non-contiguous) subsequence, any alternation $a \cdots b \cdots a \cdots b \cdots$ of length $s+2$ between two distinct symbols $a$ and $b$. The close relationship between Davenport--Schinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive, because a wide variety of geometric problems can be formulated in terms of lower envelopes. A close to linear bound on the maximum length of Davenport--Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems. This paper gives a comprehensive survey on the theory of Davenport--Schinzel sequences and their geometric applications.

http://www.math.tau.ac.il/~michas/dssurvey.pdf

Davenport-Schinzel sequences and their geometric applications

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

https://www.microsoft.com/en-us/research/uploads/prod/2016/11/Convergent-Sequences-of-Dense-Graphs-I-Subgraph-Frequencies-Metric-Properties-and-Testing.pdf

Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing

We prove the following result which is the planar version of a conjecture of Kavraki, Latombe, Motwani, and Raghavan: there is a functionf(h, ɛ) polynomial inh and 1/ɛ such that ifX is a compact planar set of Lebesgue measure 1 withh holes, such that any pointx ∈X sees a part ofX of measure at leastɛ, then there is a setG of at mostf(h, ɛ) points (guards) inX such that any point ofX is seen by at least one point ofG. With a high probability, a setG off(h, ɛ) random points inX (chosen uniformly and independently) has the above property. In the proof (givingf(h, ɛ)≤(2+o(1))1/ɛ log 1/ɛ log2
 h) we apply ideas of Kalai and Matousek who proved a weaker boundf(h, ɛ)≤C(h)1/ɛ log 1/ɛ, whereC(h) is a ‘quite fast growing function’ ofh. We improve their bound by showing a stronger result on the so-called VC-dimension of related set systems.

Guarding galleries where no point sees a small area

A {\em geometric graph\/} is a graph $G=(V,E)$ drawn in the plane so that the vertex set $V$ consists of points in general position and the edge set $E$ consists of straight line segments between points of $V$. Two edges of a geometric graph are said to be {\em parallel\/}, if they are opposite sides of a convex quadrilateral. In this paper we show that, for any fixed $k\ge3$, any geometric graph on $n$ vertices with no $k$ pairwise parallel edges contains at most $O(n)$ edges, and any geometric graph on $n$ vertices with no $k$ pairwise crossing edges contains at most $O(n\log n)$ edges. We also prove a conjecture of Kupitz that any geometric graph on $n$ vertices with no pair of parallel edges contains at most $2n-2$ edges.

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On geometric graphs with no k pairwise parallel edges

Topics in Discrete Mathematics

A subset A of a finite set P of points in the plane is called an empty polygon, if each point of A is a vertex of the convex hull of A and the convex hull of A contains no other points of P. We construct a set of n points in general position in the plane with only ˜1.62n2 empty triangles, ˜1.94n2 empty quadrilaterals, ˜1.02n2 empty pentagons, and ˜0.2n2 empty hexagons.

Planar point sets with a small number of empty convex polygons

We prove a fractional version of the Erdős—Szekeres theorem: for any k there is a constant c k > 0 such that any sufficiently large finite set X⊂R 2 contains k subsets Y 1 , ... ,Y k , each of size ≥ c k |X| , such that every set {y 1 ,...,y k } with y i e Y i is in convex position. The main tool is a lemma stating that any finite set X⊂R d contains ``large'' subsets Y 1 ,...,Y k such that all sets {y 1 ,...,y k } with y i e Y i have the same geometric (order) type. We also prove several related results (e.g., the positive fraction Radon theorem, the positive fraction Tverberg theorem). 26 June, 1998 Editors-in-Chief: la href=../edboard.html#chiefslJacob E. Goodman, Richard Pollackl/al 19n3p335.pdf yes no no yes

/pdf/a-positive-fraction-erdos-szekeres-theorem-10yn2yguay.pdf

Pavel Valtr

Papers

Guarding galleries where no point sees a small area

On geometric graphs with no k pairwise parallel edges

Topics in Discrete Mathematics

Planar point sets with a small number of empty convex polygons

A Positive Fraction Erdos - Szekeres Theorem