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.

For any sequence of positive integers j 1 < j 2 < · · · < j n , the k-tuples (j i , j i+1 , ..., j i+k−1 ), i = 1, 2, . . . , n−k+1, are said to form a monotone path of length n. Given any integers n ≥ k ≥ 2 and q ≥ 2, what is the smallest integer N with the property that no matter how we color all k-element subsets of [N ] = {1, 2, . . . , N } with q colors, we can always find a monochromatic monotone path of length n? Denoting this minimum by N k (q, n), it follows from the seminal 1935 paper of Erdős and Szekeres that N 2 (q, n) = (n − 1) q + 1 and N 3 (2, n) = 2n−4 n−2 + 1. Determining the other values of these functions appears to be a difficult task. Here we show that 2for q ≥ 2 and n ≥ q + 2. Using a "stepping-up" approach that goes back to Erdős and Hajnal, we prove analogous bounds on N k (q, n) for larger values of k, which are towers of height k − 1 in n q−1 . As a geometric application, we prove the following extension of the Happy Ending Theorem. Every family of at least M (n) = 2 n 2 log n plane convex bodies in general position, any pair of which share at most two boundary points, has n members in convex position, that is, it has n members such that each of them contributes a point to the boundary of the convex hull of their union.

A graph drawn in the plane is called k-quasi-planar if it does not contain k

In this paper, we prove several extremal results for geometrically defined hypergraphs. In particular, we establish an improved lower bound, single exponentially decreasing in k, on the best constant δ > 0 such that the vertex classes P 1 , . . . , P k of every k-partite k-uniform semi-The best previously known lower bound on δ due to Bukh and Hubard decreased triple exponentially fast in k. We give three geometric applications of our results. In particular, we establish the following strengthening of the socalled same-type lemma of Bárány and Valtr: Any disjoint finite sets P 1 , . . . ,with the property that every k-tuple formed by taking one point from each P ′ i has the same order type. We also improve a result of Fox, Gromov, Lafforgue, Naor, and Pach, who established a regularity lemma for semi-algebraic k-uniform hypergraphs of bounded complexity, showing that for each ε > 0 the vertex set can be equitably partitioned into a bounded number of parts (in terms of ε and the complexity) so that all but an ε-fraction of the k-tuples of parts are homogeneous. Here, we prove that the number of parts can be taken to be polynomial in 1/ε. Our improved regularity lemma can be applied to geometric problems and to the following general question on property testing: is it possible to decide, with query complexity polynomial in the reciprocal of the approximation parameter, whether a hypergraph has a given hereditary property? We give an affirmative answer for testing typical hereditary properties for semialgebraic hypergraphs of bounded complexity.

The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every n-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least e (log n) 1−o(1) . The dependence on the VC-dimension is hidden in the o(1) term. This improves the general lower bound, e c √ log n , due to Erdős and Hajnal, which is valid in the class of graphs satisfying any fixed nontrivial hereditary property. Our result is almost optimal and nearly matches the celebrated Erdős-Hajnal conjecture, according to which one can always find a clique or an independent set of size at least e Ω(log n) . Our results partially explain why most geometric intersection graphs arising in discrete and computational geometry have exceptionally favorable Ramsey-type properties.Our main tool is a partitioning result found by Lovász-Szegedy and Alon-Fischer-Newman, which is called the "ultra-strong regularity lemma" for graphs with bounded VC-dimension. We extend this lemma to k-uniform hypergraphs, and prove that the number of parts in the partition can be taken to be (1/ε) O(d) , improving the original bound of (1/ε) O(d 2 ) in the graph setting. We show that this bound is tight up to an absolute constant factor in the exponent. Moreover, we give an O(n k )-time algorithm for finding a partition meeting the requirements. Finally, we establish tight bounds on Ramsey-Turán numbers for graphs with bounded VC-dimension.