A bipartite graph G is semi-algebraic in R d if its vertices are represented by point sets P, Q ⊂ R d and its edges are defined as pairs of points (p, q) ∈ P × 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,kfree semi-algebraic bipartite graph G = (P, Q, E) in R 2 with |P | = m and |Q| = 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)where here ε is an arbitrarily small constant and C = C(d, k, t, ε). This result is a farreaching generalization of the classical Szemerédi-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 pointvariety incidence bound in R d , an improved bound for a d-dimensional variant of the Erdős unit distances problem, and more.

Let P 1 and P 2 be two finite sets of points in the plane, so that P 1 is contained in a line ℓ 1 , P 2 is contained in a line ℓ 2 , and ℓ 1 and ℓ 2 are neither parallel nor orthogonal. Then the number of distinct distances determined by the pairs ofIn particular, if |P 1 | = |P 2 | = m, then the number of these distinct distances is Ω(m 4/3 ), improving upon the previous bound Ω(m 5/4 ) of Elekes [3].

We study the maximal number of triangulations that a planar set of n points can have, and show that it is at most 30 n . This new bound is achieved by a careful optimization of the charging scheme of , which has led to the previous best upper bound of 43 n for the problem.Moreover, this new bound is useful for bounding the number of other types of planar (i.e., crossing-free) straight-line graphs on a given point set. Specifically, we derive new upper bounds for the number of planar graphs (O * (239.4 n )), spanning cycles (O * (70.21 n )), spanning trees (160 n ), and cycle-free graphs (O * (202.5 n )).Previous work. Variants of this problem have been studied for over 250 years. The first to consider such a variant was probably Euler, who studied the case of n points in convex position. Euler produced a recursion for the number of triangulations of such sets and guessed its solution, but could not prove its validity. In the 19th century, the problem was studied independently by several mathematicians, which were able to produce some findings, including a proof of Euler's guessed solution. That is, the number of triangulations for the convex case is C n−2 , where C m := 1 m+1 2m m = Θ(m −3/2 4 m ) = Θ * (4 m ), m ∈ N0, is the mth Catalan number 1 (see [22, page 212] for a discussion).During the mid-20th century, Tutte studied several variants of this problem, which did consider points in general position, but had other distinctions from the problem we study (see [23], and [24, pages 114-120]). Avis was perhaps one of the first to ask whether the maximum number of triangulations of n points in the plane is bounded by c n for some c > 0; see [4, page 9]. This fact was established in 1982 by Ajtai, Chvátal, Newborn, and Szemerédi [4], who show that there are at most 10 13n crossing-free graphs on n points-in particular, this bound holds for triangulations.Further developments have yielded progressively better upper bounds for the number of triangulations 2 [21,7,17], so far culminating in the previously mentioned 43 n bound [20] in 2006. This compares to Ω * (8.48 n ), the largest known number of triangulations for a set of n points, derived by Aichholzer et al. [1].The value of tr(n) has also been studied. In a companion paper [18], we derive the bound tr(n) = Ω(2.43 n ) (which improves a previous bound by Aichholzer, Hurtado, and Noy [2]). McCabe and Seidel [11] showed that when the convex hull has only O(1) vertices, there are Ω(2.63 n ) triangulations.Hurtado and Noy [9] presented a configuration of n points in general position and Θ * ( √ 12 n ) ≈

We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, nonweighted common geometric graphs drawn on n points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of n points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits Ω(8.65 n ) different triangulations. This improves the bound Ω(8.48 n ) achieved by the previous best construction, the double zig-zag chain studied by Aichholzer et al. (ii) We obtain a new lower bound of Ω(12.00 n ) for the number of noncrossing spanning trees of the double chain composed of two convex chains. The previous bound, Ω(10.42 n ), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30 n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.62 n ) noncrossing spanning cycles. (iv) We derive lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). We show that the number of shortest tours can be exponential in n for points in general position. These tours are automatically noncrossing. Likewise, we show that the number of longest noncrossing tours can be exponential in n. It was known that the number of shortest noncrossing perfect matchings can be exponential in n, and here we show that the number of longest noncrossing perfect matchings can be also exponential in n. It was known that the number of longest noncrossing spanning trees of a point set can be exponentially large, and here we show that this can be also realized with points in convex position. For points in convex position we re-derive tight bounds for the number of longest and shortest tours with some simpler arguments. We also give a combinatorial characterization of longest tours, which yields an O(n log n) time algorithm for computing them.

We generalize the notions of flippable and simultaneously flippable edges in a tri-angulation of a set S of points in the plane to so-called pseudo-simultaneously flippable edges. Such edges are related to the notion of convex decompositions spanned by S. We prove a worst-case tight lower bound for the number of pseudo-simultaneously flippable edges in a triangulation in terms of the number of vertices. We use this bound for deriving new upper bounds for the maximal number of crossing-free straight-edge graphs that can be embedded on any fixed set of N points in the plane. We obtain new upper bounds for the number of spanning trees and forests as well. Specifically, let tr(N) denote the maximum number of triangulations on a set of N points in the plane. Then we show (using the known bound tr(N) < 30 N) that any N-element point set admits at most 6.9283 N · tr(N) < 207.85 N crossing-free straight-edge graphs, O(4.7022 N) · tr(N) = O(141.07 N) spanning trees, and O(5.3514 N) · tr(N) = O(160.55 N) forests. We also obtain upper bounds for the number of crossing-free straight-edge graphs that have cN , fewer than cN , or more than cN edges, for any constant parameter c, in terms of c and N .

The local properties problem of Erdős and Shelah generalizes many Ramsey problems and some distinct distances problems. In this work, we derive a variety of new bounds for the local properties problem and its variants. We do this by continuing to develop the color energy technique -a variant of the concept of additive energy from Additive Combinatorics. In particular, we generalize the concept of color energy to higher color energies, and combine these with Extremal Graph Theory results about graphs with no cycles or subdivisions of size k. *

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We establish an improved upper bound for the number of incidences between m points and n circles in three dimensions. The previous best known bound, originally established for the planar case and later extended to any dimension ≥ 2, is O * m 2/3 n 2/3 + m 6/11 n 9/11 + m + n , where the O * (·) notation hides polylogarithmic factors. Since all the points and circles may lie on a common plane (or sphere), it is impossible to improve the bound in R 3 without first improving it in the plane.Nevertheless, we show that if the set of circles is required to be "truly threedimensional" in the sense that no sphere or plane contains more than q of the circles, for some q ≪ n, then for any ε > 0 the bound can be improved to O m 3/7+ε n 6/7 + m 2/3+ε n 1/2 q 1/6 + m 6/11+ε n 15/22 q 3/22 + m + n .For various ranges of parameters (e.g., when m = Θ(n) and q = o(n 7/9 )), this bound is smaller than the lower bound Ω * (m 2/3 n 2/3 + m + n), which holds in two dimensions.We present several extensions and applications of the new bound: (i) For the special case where all the circles have the same radius, we obtain the improved bound O m 5/11+ε n 9/11 + m 2/3+ε n 1/2 q 1/6 + m + n . (ii) We present an improved analysis that removes the subpolynomial factors from the bound when m = O(n 3/2−ε ) for any fixed ε > 0. (iii) We use our results to obtain the improved bound O(m 15/7 ) for the number of mutually similar triangles determined by any set of m points in R 3 .Our result is obtained by applying the polynomial partitioning technique of Guth and Katz using a constant-degree partitioning polynomial (as was also recently used by Solymosi and Tao). We also rely on various additional tools from analytic, algebraic, and combinatorial geometry.