General Position Subsets and Independent Hyperplanes in d-Space

Erd\H{o}s asked what is the maximum number $\alpha(n)$ such that every set of $n$ points in the plane with no four on a line contains $\alpha(n)$ points in general position. We consider variants of this question for $d$-dimensional point sets and generalize previously known bounds. In particular, we prove the following two results for fixed $d$: - Every set $H$ of $n$ hyperplanes in $\mathbb{R}^d$ contains a subset $S\subseteq H$ of size at least $c \left(n \log n\right)^{1/d}$, for some constant $c=c(d)>0$, such that no cell of the arrangement of $H$ is bounded by hyperplanes of $S$ only. - Every set of $cq^d\log q$ points in $\mathbb{R}^d$, for some constant $c=c(d)>0$, contains a subset of $q$ cohyperplanar points or $q$ points in general position. Two-dimensional versions of the above results were respectively proved by Ackerman et al. [Electronic J. Combinatorics, 2014] and by Payne and Wood [SIAM J. Discrete Math., 2013].Comment: 8 page

Számelmélet és kombinatorikus vonatkozásai = Number Theory and its Interactions with Combinatorics

A kutatók számos érdekes eredményt értek el a kombinatorikus számelmélet és geometria, gráfelmélet, diofantikus approximáció területén, itt csak néhányat említünk. Elekes és Ruzsa a Freiman, Balog-Szemerédi és Laczkovich-Ruzsa tételek közös általánosítását adják, ezzel a témakört egységesítik, és számos kombinatorikus geometriai tételt fejlesztenek tovább. Elekes Szabó E.-vel áttörést ért el a sok szabályosságot tartalmazó konfigurációk karakterizációjának általános problémájában, néhány korábbi eredményt jelentősen továbbfejlesztve. Szemerédi A. Khalfalah-val igazolja Sárközy, Roth és T. Sós azon sejtését, hogy: ha beosztjuk az egész számokat véges sok osztályba, akkor valamely osztályban van két olyan szám, amelyek összege négyzetszám, V. Vu-val közösen pedig Folkman egy sejtését bizonyítja. Biró javítja Ruzsa és Kolountzakis egész számok parkettázására vonatkozó eredményét. Erősíti és általánosítja a "karakterizáló sorozatok" témakör korábbi eredményeit. Ruzsa és B. Green meghatározzák tetszőleges véges kommutatív csoportban a legnagyobb összegmentes halmaz elemszámát. T. Sós Lovász L.-val megmutatja, hogy ha gráfok egy sorozatában a kis részgráfoknak ugyanaz az eloszlása, mint egy általánosított G véletlen gráfban, akkor ezen gráfoknak aszimptotikusan olyan struktúrája van, mint G-nek. T. Sós társszerzőkkel azt az alapkérdést vizsgálja, mikor van közel egymáshoz két gráf. | The participants obtaind several interesting results in combinatorial number theory and geometry, graph theory, diophantine approximation, we list just a few of these results.. Elekes and Ruzsa give a common generalization of the Freiman, Balog-Szemerédi and Laczkovich-Ruzsa theorems, unifying in this way the subject and improving a lot of earlier results. Elekes with E. Szabó achieved a breakthrough in the general problem of characterizing configurations having a lot of reguarity, improving some earlier results. Szemerédi with A. Khalfalah proves the follwing conjecture of Sárközy, Roth and T. Sós: if we divide the set of integers into finitely many classes, then in one of the classes we can find two numbers such that their sum is a square, and with V. Vu he proves a conjecture of Folkman. Biró improves a result of Ruzsa and Kolountzakis on tilings of the integers, and, he proves generalizations and strengthenings of some results in the subject 'characterizing sequences'. Ruzsa and B. Green determine the size of the largest sumfree set in an arbitrary finite Abelian group. L. Lovász and T. Sós showed that generalized quasirandom sequences (whose subgraph densities match those of a fixed finite weighted graph) have a finite structure. T. Sós with co-authors defines the distance of two graphs that reflects the similarity , the closeness of both local and global properties

On distinct distances in homogeneous sets in the Euclidean space

A homogeneous set of $n$ points in the $d$-dimensional Euclidean space determines at least $\Omega(n^{2d/(d^2+1)} / \log^{c(d)} n)$ distinct distances for a constant $c(d)>0$. In three-space, we slightly improve our general bound and show that a homogeneous set of $n$ points determines at least $\Omega(n^{.6091})$ distinct distances

A Bichromatic Incidence Bound and an Application

We prove a new, tight upper bound on the number of incidences between points and hyperplanes in Euclidean d-space. Given n points, of which k are colored red, there are O_d(m^{2/3}k^{2/3}n^{(d-2)/3} + kn^{d-2} + m) incidences between the k red points and m hyperplanes spanned by all n points provided that m = \Omega(n^{d-2}). For the monochromatic case k = n, this was proved by Agarwal and Aronov. We use this incidence bound to prove that a set of n points, no more than n-k of which lie on any plane or two lines, spans \Omega(nk^2) planes. We also provide an infinite family of counterexamples to a conjecture of Purdy's on the number of hyperplanes spanned by a set of points in dimensions higher than 3, and present new conjectures not subject to the counterexample.Comment: 12 page

An improved bound on the number of point-surface incidences in three dimensions

We show that $m$ points and $n$ smooth algebraic surfaces of bounded degree in $\mathbb{R}^3$ satisfying suitable nondegeneracy conditions can have at most $O(m^{\frac{2k}{3k-1}}n^{\frac{3k-3}{3k-1}}+m+n)$ incidences, provided that any collection of $k$ points have at most O(1) surfaces passing through all of them, for some $k\geq 3$. In the case where the surfaces are spheres and no three spheres meet in a common circle, this implies there are $O((mn)^{3/4} + m +n)$ point-sphere incidences. This is a slight improvement over the previous bound of $O((mn)^{3/4} \beta(m,n)+ m +n)$ for $\beta(m,n)$ an (explicit) very slowly growing function. We obtain this bound by using the discrete polynomial ham sandwich theorem to cut $\mathbb{R}^3$ into open cells adapted to the set of points, and within each cell of the decomposition we apply a Turan-type theorem to obtain crude control on the number of point-surface incidences. We then perform a second polynomial ham sandwich decomposition on the irreducible components of the variety defined by the first decomposition. As an application, we obtain a new bound on the maximum number of unit distances amongst $m$ points in $\mathbb{R}^3$.Comment: 17 pages, revised based on referee comment