66,783 research outputs found
Ramsey-type theorems for lines in 3-space
We prove geometric Ramsey-type statements on collections of lines in 3-space.
These statements give guarantees on the size of a clique or an independent set
in (hyper)graphs induced by incidence relations between lines, points, and
reguli in 3-space. Among other things, we prove that: (1) The intersection
graph of n lines in R^3 has a clique or independent set of size Omega(n^{1/3}).
(2) Every set of n lines in R^3 has a subset of n^{1/2} lines that are all
stabbed by one line, or a subset of Omega((n/log n)^{1/5}) such that no
6-subset is stabbed by one line. (3) Every set of n lines in general position
in R^3 has a subset of Omega(n^{2/3}) lines that all lie on a regulus, or a
subset of Omega(n^{1/3}) lines such that no 4-subset is contained in a regulus.
The proofs of these statements all follow from geometric incidence bounds --
such as the Guth-Katz bound on point-line incidences in R^3 -- combined with
Tur\'an-type results on independent sets in sparse graphs and hypergraphs.
Although similar Ramsey-type statements can be proved using existing generic
algebraic frameworks, the lower bounds we get are much larger than what can be
obtained with these methods. The proofs directly yield polynomial-time
algorithms for finding subsets of the claimed size.Comment: 18 pages including appendi
Point-plane incidences and some applications in positive characteristic
The point-plane incidence theorem states that the number of incidences
between points and planes in the projective three-space over a
field , is where is the maximum number
of collinear points, with the extra condition if has
characteristic . This theorem also underlies a state-of-the-art
Szemer\'edi-Trotter type bound for point-line incidences in , due to
Stevens and de Zeeuw.
This review focuses on some recent, as well as new, applications of these
bounds that lead to progress in several open geometric questions in , for
. These are the problem of the minimum number of distinct nonzero
values of a non-degenerate bilinear form on a point set in , the analogue
of the Erd\H os distinct distance problem in and additive energy
estimates for sets, supported on a paraboloid and sphere in . It avoids
discussing sum-product type problems (corresponding to the special case of
incidences with Cartesian products), which have lately received more attention.Comment: A survey, with some new results, for the forthcoming Workshop on
Pseudorandomness and Finite Fields in at RICAM in Linz 15-19 October, 2018;
24p
Sphere tangencies, line incidences, and Lie's line-sphere correspondence
Two spheres with centers and and signed radii and are said to
be in contact if . Using Lie's line-sphere correspondence,
we show that if is a field in which is not a square, then there is an
isomorphism between the set of spheres in and the set of lines in a
suitably constructed Heisenberg group that is embedded in ; under
this isomorphism, contact between spheres translates to incidences between
lines.
In the past decade there has been significant progress in understanding the
incidence geometry of lines in three space. The contact-incidence isomorphism
allows us to translate statements about the incidence geometry of lines into
statements about the contact geometry of spheres. This leads to new bounds for
Erd\H{o}s' repeated distances problem in , and improved bounds for the
number of point-sphere incidences in three dimensions. These new bounds are
sharp for certain ranges of parameters.Comment: 20 pages, 2 figures. v2: minor changes in response to referee
comments. To appear in Math. Proc. Camb. Philos. So
On the number of tetrahedra with minimum, unit, and distinct volumes in three-space
We formulate and give partial answers to several combinatorial problems on
volumes of simplices determined by points in 3-space, and in general in
dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by
points in \RR^3 is at most , and there are point sets
for which this number is . We also present an time
algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby
extend an algorithm of Edelsbrunner, O'Rourke, and Seidel. In general, for
every k,d\in \NN, , the maximum number of -dimensional
simplices of minimum (nonzero) volume spanned by points in \RR^d is
. (ii) The number of unit-volume tetrahedra determined by
points in \RR^3 is , and there are point sets for which this
number is . (iii) For every d\in \NN, the minimum
number of distinct volumes of all full-dimensional simplices determined by
points in \RR^d, not all on a hyperplane, is .Comment: 19 pages, 3 figures, a preliminary version has appeard in proceedings
of the ACM-SIAM Symposium on Discrete Algorithms, 200
Incidences between points and lines in three dimensions
We give a fairly elementary and simple proof that shows that the number of
incidences between points and lines in , so that no
plane contains more than lines, is (in the precise statement, the constant
of proportionality of the first and third terms depends, in a rather weak
manner, on the relation between and ).
This bound, originally obtained by Guth and Katz~\cite{GK2} as a major step
in their solution of Erd{\H o}s's distinct distances problem, is also a major
new result in incidence geometry, an area that has picked up considerable
momentum in the past six years. Its original proof uses fairly involved
machinery from algebraic and differential geometry, so it is highly desirable
to simplify the proof, in the interest of better understanding the geometric
structure of the problem, and providing new tools for tackling similar
problems. This has recently been undertaken by Guth~\cite{Gu14}. The present
paper presents a different and simpler derivation, with better bounds than
those in \cite{Gu14}, and without the restrictive assumptions made there. Our
result has a potential for applications to other incidence problems in higher
dimensions
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