arXiv subjects
Publications and source records attributed to Jozsef Solymosi.
We prove almost tight bounds on incidences between points and $k$-dimensional varieties of bounded degree in $\R^d$. Our main tools are the Polynomial Ham Sandwich Theorem and induction on both the dimension and the number of points.
We establish new bounds on the number of tangencies and orthogonal intersections determined by an arrangement of curves. First, given a set of $n$ algebraic plane curves, we show that there are $O(n^{3/2})$ points where two or more curves are tangent. In particular, if no three curves are mutually tangent at a common point, then there are $O(n^{3/2})$ curve-curve tangencies. Second, given a family of algebraic plane curves and a set of $n$ curves from this family, we show that either there are $O(n^{3/2})$ points where two or more curves are orthogonal, or the family of curves has certain special properties. We obtain these bounds by transforming the arrangement of plane curves into an arrangement of space curves so that tangency (or orthogonality) of the original plane curves corresponds to intersection of space curves. We then bound the number of intersections of the corresponding space curves. For the case of curve-curve tangency, we use a polynomial method technique that is reminiscent of Guth and Katz's proof of the joints theorem. For the case of orthogonal curve intersections, we employ a bound of Guth and the third author to control the number of two-rich points in space curve arrangements.
The main result of this paper is that for any $c>0$ and for large enough $n$ if the number of edges in a 3-uniform hypergraph is at least $cn^2$ then there is a core (subgraph with minimum degree at least 2) on at most 15 vertices. We conjecture that our result is not sharp and 15 can be replaced by 9. Such an improvement seems to be out of reach, since it would imply the following case of a long-standing conjecture by Brown, Erd\H os, and Sós; if there is no set of 9 vertices that span at least 6 edges of a 3-uniform hypergraph then it is sparse.
We give bounds on the number of distinct differences $N_a-a$ as $a$ varies over all elements of a given finite set $A$, and $N_a$ is a nearest neighbour to $a$.
The Subspace Theorem is a powerful tool in number theory. It has appeared in various forms and been adapted and improved over time. It's applications include diophantine approximation, results about integral points on algebraic curves and the construction of transcendental numbers. But its usefulness extends beyond the realms of number theory. Other applications of the Subspace Theorem include linear recurrence sequences and finite automata. In fact, these structures are closely related to each other and the construction of transcendental numbers. The Subspace Theorem also has a number of remarkable combinatorial applications. The purpose of this paper is to give a survey of some of these applications including sum-product estimates and bounds on unit distances. The presentation will be from the point of view of a discrete mathematician. We will state a number of variants of the Subspace Theorem below but we will not prove any of them as the proofs are beyond the scope of this work. However we will give a proof of a simplified special case of the Subspace Theorem which is still very useful for many problems in discrete mathematics.
The first open case of the Brown, Erdős, Sós conjecture is equivalent to the following; For every $c>0$ there is a threshold $n_0$ so that if a quasigroup has order $n\geq n_0$ then for every subset of triples of the form $(a,b,ab),$ denoted by $S,$ if $|S|\geq cn^2$ then there is a seven-element subset of the quasigroup which spans at least four triples of the selected subset $S.$ In this paper we prove the conjecture for finite groups.
Let $p_1,p_2,p_3$ be three non-collinear points in the plane, and let $P$ be a set of $n$ other points in the plane. We show that the number of distinct distances between $p_1,p_2,p_3$ and the points of $P$ is $Ω(n^{6/11})$, improving the lower bound $Ω(n^{0.502})$ of Elekes and Szabó \cite{ESz} (and considerably simplifying the analysis).
Let $F(x,y)$ be a polynomial over the rationals. We show that if $F$ is not an expander (over the rationals) then it has a special multiplicative or additive form. For example if $F$ is a homogeneous non-expander polynomial then $F(x,y)=c(x+ay)^α$ or $F(x,y)=c(xy)^α.$ This is an extension of an earlier result of Elekes and Rónyai who described the structure of two-variate polynomials which are not expanders over the reals.
We prove Roth type theorems in finite groups. Our main tool is the Triangle Removal Lemma of Ruzsa and Szemerédi.
Ulam asked in 1945 if there is an everywhere dense \emph{rational set}, i.e. a point set in the plane with all its pairwise distances rational. Erd\H os conjectured that if a set $S$ has a dense rational subset, then $S$ should be very special. The only known types of examples of sets with dense (or even just infinite) rational subsets are lines and circles. In this paper we prove Erd\H os's conjecture for algebraic curves, by showing that no irreducible algebraic curve other than a line or a circle contains an infinite rational set.
In this note we establish a Ramsey-type result for certain subsets of the $n$-dimensional cube. This can then be applied to obtain reasonable bounds on various related structures, such as (partial) Hales-Jewett lines for alphabets of sized 3 and 4, Hilbert cubes in sets of real numbers with small sumsets, "corner" in the integer lattice in the plane, and 3-term geometric progressions in integers.
We prove that the sumset or the productset of any finite set of real numbers, $A,$ is at least $|A|^{4/3-ε},$ improving earlier bounds. Our main tool is a new upper bound on the multiplicative energy, $E(A,A).$
We present some elementary ideas to prove the following Sylvester-Gallai type theorems involving incidences between points and lines in the planes over the complex numbers and quaternions. (1) Let A and B be finite sets of at least two complex numbers each. Then there exists a line l in the complex affine plane such that l intersects AxB in exactly two points. (2) Let S be a finite noncollinear set of points in the complex affine plane. Then there exists a line l intersecting S in 2, 3, 4 or 5 points. (3) Let A and B be finite sets of at least two quaternions each. Then there exists a line l in the quaternionic affine plane such that l intersects AxB in 2, 3, 4 or 5 points. (4) Let S be a finite noncollinear set of points in the quaternionic affine plane. Then there exists a line l intersecting S in at least 2 and at most 24 points.
Given a set of $s$ points and a set of $n^2$ lines in three-dimensional Euclidean space such that each line is incident the $n$ points but no $n$ lines are coplanar, then we have $s=Ω(n^{11/4})$. This is the first nontrivial answer to a question recently posed by Jean Bourgain.
We prove Pach-Sharir type incidence theorems for a class of curves in R^n and surfaces in R^3, which we call pseudoflats. In particular, our results apply to a wide class of generic irreducible real algebraic sets of bounded degree.
The Szemerédi-Trotter theorem gives a bound on the maximum number of incidences between points and lines on the Euclidean plane. In particular it says that $n$ lines and $n$ points determine $O(n^{4/3})$ incidences. Let us suppose that an arrangement of $n$ lines and $n$ points defines $cn^{4/3}$ incidences, for a given positive $c.$ It is widely believed that such arrangements have special structure, but no results are known in this direction. Here we show that for any natural number, $k,$ one can find $k$ points of the arrangement in general position such that any pair of them is incident to a line from the arrangement, provided by $n\geq n_0(k).$ In a subsequent paper we will establish similar statement to hyperplanes.
We present an elementary proof that if $A$ is a finite set of numbers, and the sumset $A+_GA$ is small, $|A+_GA|\leq c|A|$, along a dense graph $G$, then $A$ contains $k$-term arithmetic progressions.