A semi-algebraic version of Zarankiewicz's problem

A bipartite graph G is semi-algebraic in R-d if its vertices are represented by point sets P,Q subset of R-d and its edges are defined as pairs of points (p,q) epsilon P x Q that satisfy a Boolean combination of a fixed number of polynomial equations and inequalities in 2d coordinates. We show that for fixed k, the maximum number of edges in a K-k,K-k-free semi-algebraic bipartite graph G = (P, Q E) in R-2 with vertical bar P vertical bar = m and vertical bar Q vertical bar = n is at most O ((mn)(2/3) + m + n), and this bound is tight. In dimensions d >= 3, we show that all such semi-algebraic graphs have at most C((mn)(d/(d+1)+epsilon) + m + n) edges, where epsilon is an arbitrarily small constant and C = C(d, k, t, epsilon). This result is a far-reaching generalization of the classical Szemeredi-Trotter incidence theorem. The proof combines tools from several fields: VC-dimension and shatter functions, polynomial partitioning, and Hilbert polynomials. We also present various applications of our theorem, for example, a general point-variety incidence bound in R-d, an improved bound for a d-dimensional variant of the Erdos unit distances problem, and more.

Ramsey-type results for semi-algebraic relations

János Pach

A k-ary semi-algebraic relation E on R^d is a subset of R^{kd}, the set of k-tuples of points in R^d, which is determined by a finite number of polynomial equations and inequalities in kd real variables. The description complexity of such a relation is at most t if the number of polynomials and their degrees are all bounded by t. A subset A of R^d is called homogeneous if all or none of the k-tuples from A satisfy E. A large number of geometric Ramsey-type problems and results can be formulated as questions about finding large homogeneous subsets of sets in R^d equipped with semi-algebraic relations. In this paper we study Ramsey numbers for k-ary semi-algebraic relations of bounded complexity and give matching upper and lower bounds, showing that they grow as a tower of height k-1. This improves on a direct application of Ramsey's theorem by one exponential and extends a result of Alon, Pach, Pinchasi, Radoi\v{c}i'c, and Sharir, who proved this for k=2. We apply our results to obtain new estimates for some geometric Ramsey-type problems relating to order types and one-sided sets of hyperplanes. We also study the off-diagonal case, achieving some partial results.

2013

On some algebraic and extremal problems in discrete geometry

Seyed Hossein Nassajianmojarrad

In the present thesis, we delve into different extremal and algebraic problems arising from combinatorial geometry. Specifically, we consider the following problems. For any integer

n\ge 3

, we define

e(n)

to be the minimum positive integer such that any set of

e(n)

points in general position in the plane contains

n

points in convex position. In 1935, Erd\H{o}s and Szekeres proved that

e(n) \le {2n-4 \choose n-2}+1

and later in 1961, they obtained the lower bound

2^{n-2}+1 \le e(n)

, which they conjectured to be optimal. We prove that

e(n) \le {2n-5 \choose n-2}-{2n-8 \choose n-3}+2

. In a recent breakthrough, Suk proved that

e(n) \le 2^{n+O\left(n^{2/3}\log n\right)}

. We strengthen this result by extending it to pseudo-configurations and also improving the error term. Combining our results with a theorem of Dobbins et al., we significantly improve the best known upper bounds on the following two functions, introduced by Bisztriczky and Fejes T'{o}th and by Pach and T'{o}th, respectively. Let

c(n)

(and

c'(n)

) denote the smallest positive integer

N

such that any family of

N

pairwise disjoint convex bodies in general position (resp.,

N

convex bodies in general position, any pair of which share at most two boundary points) has an

n

members in convex position. We show that

c(n)\le c'(n)\le 2^{n+O\left(\sqrt{n\log n}\right)}

. Given a point set

P

in the plane, an ordinary circle for

P

is defined as a circle containing exactly three points of

P

. We prove that any set of

n

points in the plane, not all on a line or a circle, determines at least

\frac{1}{4}n^2-O(n)

ordinary circles. We determine the exact minimum number of ordinary circles for all sufficiently large

n

, and characterize all point sets that come close to this minimum. We also consider the orchard problem for circles, where we determine the maximum number of circles containing four points of a given set and describe the extremal configurations. A special case of the Schwartz-Zippel lemma states that given an algebraic curve

C\subset \mathbb{C}^2

of degree

d

and two finite sets

A,B\subset \mathbb{C}

, we have

|C\cap (A\times B)|=O_d(|A|+|B|)

. We establish a two-dimensional version of this result, and prove upper bounds on the size of the intersection

|X\cap (P\times Q)|

for a variety

X\subset \mathbb{C}^4

and finite sets

P,Q\subset \mathbb{C}^2

. A key ingredient in our proofs is a two-dimensional version of a special case of Alon's combinatorial Nullstellensatz. As corollaries, we generalize the Szemer'edi-Trotter point-line incidence theorem and several known bounds on repeated and distinct Euclidean distances. We use incidence geometry to prove some sum-product bounds over arbitrary fields. First, we give an explicit exponent and improve a recent result of Bukh and Tsimerman by proving that

\max \{ |A+A|, |f(A, A)|\}\gg |A|^{6/5}

for any small set

A\subset \mathbb{F}_p

and quadratic non-degenerate polynomial

f(x, y)\in \mathbb{F}_p[x, y]

. This generalizes the result of Roche-Newton et al. giving the best known lower bound for the term

\max \{ |A+A|, |A \cdot A |\}

. Secondly, we improve and generalize the sum-product results of Hegyv'{a}ri and Hennecart on

\max\{ |A+B|, |f(B,C)|\}

, for a specific type of function

f

. Finally, we prove that the number of distinct cubic distances generated by any small set

A\times A\subset \mathbb{F}_p^2

\Omega(|A|^{8/7})

, which improves a result of Yazici et al.

EPFL2017