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For every positive integer $n$, we construct a Hasse diagram with $n$ vertices and chromatic number $\Omega(n^{1/4})$, which significantly improves on the previously known best constructions of Hasse diagrams having chromatic number $\Theta(\log n)$. In addition, if we also require that our Hasse diagram has girth at least $k\geq 5$, we can achieve a chromatic number of at least $n^{\frac{1}{2k-3}+o(1)}$. These results have the following surprising geometric consequence. They imply the existence of a family $\mathcal{C}$ of $n$ curves in the plane such that the disjointness graph $G$ of $\mathcal{C}$ is triangle-free (or have high girth), but the chromatic number of $G$ is polynomial in $n$. Again, the previously known best construction, due to Pach, Tardos and T\'oth, had only logarithmic chromatic number., Comment: 11 pages, 1 figure

We consider m-colorings of the edges of a complete graph, where each color class is defined semi-algebraically with bounded complexity. The case m = 2 was first studied by Alon et al., who applied this framework to obtain surprisingly strong Ramsey-type results for intersection graphs of geometric objects and for other graphs arising in computational geometry. Considering larger values of m is relevant, e.g., to problems concerning the number of distinct distances determined by a point set. For p ≥ 3 and m ≥ 2, the classical Ramsey number R(p; m) is the smallest positive integer n such that any m-coloring of the edges of Kn, the complete graph on n vertices, contains a monochromatic Kp. It is a longstanding open problem that goes back to Schur (1916) to decide whether R(p; m) ≤ 2cm, where c = c(p). We prove that this is true if each color class is defined semi-algebraically with bounded complexity, and that the order of magnitude of this bound is tight. Our proof is based on the Cutting Lemma of Chazelle et al., and on a Szemeredi-type regularity lemma for multicolored semi-algebraic graphs, which is of independent interest. The same technique is used to address the semi-algebraic variant of a more general Ramsey-type problem of Erdős and Shelah.

Motivated by the Erdős–Szekeres convex polytope conjecture in , we initiate the study of the following induced Ramsey problem for hypergraphs. Given integers , what is the minimum integer such that any -uniform hypergraph on vertices with the property that any set of vertices induces 0, 2, or 4 edges, contains an independent set of size . Our main result shows that , where .

In 1916, Schur introduced the Ramsey number r(3;m), which is the minimum integer n > 1 such that for any m-coloring of the edges of the complete graph K_n, there is a monochromatic copy of K₃. He showed that r(3;m) ≤ O(m!), and a simple construction demonstrates that r(3;m) ≥ 2^Ω(m). An old conjecture of Erdős states that r(3;m) = 2^Θ(m). In this note, we prove the conjecture for m-colorings with bounded VC-dimension, that is, for m-colorings with the property that the set system induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension., LIPIcs, Vol. 164, 36th International Symposium on Computational Geometry (SoCG 2020), pages 46:1-46:8

The classical hypergraph Ramsey number rk(s, n) is the minimum N such that for every red-blue coloring of the k-tuples of {1, …, N}, there are s integers such that every k-tuple among them is red, or n integers such that every k-tuple among them is blue. We survey a variety of problems and results in hypergraph Ramsey theory that have grown out of understanding the quantitative aspects of rk(s, n). Our focus is on recent developments and open problems.

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, $$e^{c\sqrt{\log n}}$$ , 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 $$e^{\Omega (\log n)}$$ . 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 Lovasz–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 $$(1/\varepsilon )^{O(d)}$$ , improving the original bound of $$(1/\varepsilon )^{O(d^2)}$$ 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–Turan numbers for graphs with bounded VC-dimension.

We establish the following result related to Erdős’s problem on distinct distances. Let V be an n-element planar point set such that any p members of V determine at least \(\left( {\begin{array}{*{20}{c}} p \\ 2 \end{array}} \right) - p + 6\) distinct distances. Then V determines at least \(n^{\tfrac{8} {7} - o(1)}\) distinct distances, as n tends to infinity.

The famous Szemer\'{e}di-Trotter theorem states that any arrangement of $n$ points and $n$ lines in the plane determines $O(n^{4/3})$ incidences, and this bound is tight. In this paper, we prove the following Tur\'an-type result for point-line incidence. Let $\mathcal{L}_1$ and $\mathcal{L}_2$ be two sets of $t$ lines in the plane and let $P=\{\ell_1 \cap \ell_2 : \ell_1 \in \mathcal{L}_1, \ell_2 \in \mathcal{L}_2\}$ be the set of intersection points between $\mathcal{L}_1$ and $\mathcal{L}_2$. We say that $(P, \mathcal{L}_1 \cup \mathcal{L}_2)$ forms a \emph{natural $t\times t$ grid} if $|P| =t^2$, and $conv(P)$ does not contain the intersection point of some two lines in $\mathcal{L}_i,$ for $i = 1,2.$ For fixed $t > 1$, we show that any arrangement of $n$ points and $n$ lines in the plane that does not contain a natural $t\times t$ grid determines $O(n^{\frac{4}{3}- \varepsilon})$ incidences, where $\varepsilon = \varepsilon(t)$. We also provide a construction of $n$ points and $n$ lines in the plane that does not contain a natural $2 \times 2$ grid and determines at least $\Omega({n^{1+\frac{1}{14}}})$ incidences., Comment: 13 pages, 5 figures

Two sets $A$ and $B$ of points in the plane are \emph{mutually avoiding} if no line generated by any two points in $A$ intersects the convex hull of $B$, and vice versa. In 1994, Aronov, Erd\H os, Goddard, Kleitman, Klugerman, Pach, and Schulman showed that every set of $n$ points in the plane in general position contains a pair of mutually avoiding sets each of size at least $\sqrt{n/12}$. As a corollary, their result implies that for every set of $n$ points in the plane in general position one can find at least $\sqrt{n/12}$ segments, each joining two of the points, such that these segments are pairwise crossing. In this note, we prove a fractional version of their theorem: for every $k > 0$ there is a constant $\varepsilon_k > 0$ such that any sufficiently large point set $P$ in the plane contains $2k$ subsets $A_1,\ldots, A_{k},B_1,\ldots, B_k$, each of size at least $\varepsilon_k|P|$, such that every pair of sets $A = \{a_1,\ldots, a_k\}$ and $B = \{b_1,\ldots, b_k\}$, with $a_i \in A_i$ and $b_i \in B_i$, are mutually avoiding. Moreover, we show that $\varepsilon_k = ��(1/k^4)$. Similar results are obtained in higher dimensions, 10 pages, 2 figures

Let $\mathbb{F}_q$ be a finite field of order $q$ and $\mathcal{E}$ be a set in $\mathbb{F}_q^d$. The distance set of $\mathcal{E}$, denoted by $\Delta(\mathcal{E})$, is the set of distinct distances determined by the pairs of points in $\mathcal{E}$. Very recently, Iosevich, Koh, and Parshall (2018) proved that if $|\mathcal{E}|\gg q^{d/2}$, then the quotient set of $\Delta(\mathcal{E})$ satisfies \[\left\vert\frac{\Delta(\mathcal{E})}{\Delta(\mathcal{E})}\right\vert=\left\vert \left\lbrace\frac{a}{b}\colon a, b\in \Delta(\mathcal{E}), b\ne 0\right\rbrace\right\vert\gg q.\] In this paper, we break the exponent $d/2$ when $\mathcal{E}$ is a Cartesian product of sets over a prime field. More precisely, let $p$ be a prime and $A\subset \mathbb{F}_p$. If $\mathcal{E}=A^d\subset \mathbb{F}_p^d$ and $|\mathcal{E}|\gg p^{\frac{d}{2}-\varepsilon}$ for some $\varepsilon>0$, then we have \[\left\vert\frac{\Delta(\mathcal{E})}{\Delta(\mathcal{E})}\right\vert, ~\left\vert \Delta(\mathcal{E})\cdot \Delta(\mathcal{E})\right\vert \gg p.\] Such improvements are not possible over arbitrary finite fields. These results give us a better understanding about the structure of distance sets and the Erd\H{o}s-Falconer distance conjecture over finite fields., Comment: 8 pages. Submitted for publication

A topological graph is $k$-quasi-planar if it does not contain $k$ pairwise crossing edges. A 20-year-old conjecture asserts that for every fixed $k$, the maximum number of edges in a $k$-quasi-planar graph on $n$ vertices is $O(n)$. Fox and Pach showed that every $k$-quasi-planar graph with $n$ vertices has at most $n(\log n)^{O(\log k)}$ edges. We improve this upper bound to $2^{\alpha(n)^c}n\log n$, where $\alpha(n)$ denotes the inverse Ackermann function and $c$ depends only on $k$, for $k$-quasi-planar graphs in which any two edges intersect in a bounded number of points. We also show that every $k$-quasi-planar graph with $n$ vertices in which any two edges have at most one point in common has at most $O(n\log n)$ edges. This improves the previously known upper bound of $2^{\alpha(n)^c}n\log n$ obtained by Fox, Pach, and Suk., Comment: Final version, minor corrections

The Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there are $s$ integers such that every $k$-tuple among them is red, or $n$ integers such that every $k$-tuple among them is blue. We prove the following new lower bounds for 4-uniform hypergraph Ramsey numbers: $$r_4(5,n) > 2^{n^{c\log n}} \qquad \hbox{ and } \qquad r_4(6,n) > 2^{2^{cn^{1/5}}},$$ where $c$ is an absolute positive constant. This substantially improves the previous best bounds of $2^{n^{c\log\log n}}$ and $2^{n^{c\log n}}$, respectively. Using previously known upper bounds, our result implies that the growth rate of $r_4(6,n)$ is double exponential in a power of $n$. As a consequence, we obtain similar bounds for the $k$-uniform Ramsey numbers $r_k(k+1, n)$ and $r_k(k+2, n)$ where the exponent is replaced by an appropriate tower function. This almost solves the question of determining the tower growth rate for {\emph {all}} classical off-diagonal hypergraph Ramsey numbers, a question first posed by Erd\H os and Hajnal in 1972. The only problem that remains is to prove that $r_4(5,n)$ is double exponential in a power of $n$.

Let n>=@?>=2 and q>=2. We consider the minimum N such that whenever we have N points in the plane in general position and the @?-subsets of these points are colored with q colors, there is a subset S of n points all of whose @?-subsets have the same color and furthermore S is in convex position. This combines two classical areas of intense study over the last 75 years: the Ramsey problem for hypergraphs and the Erdos-Szekeres theorem on convex configurations in the plane. For the special case @?=2, we establish a single exponential bound on the minimum N, such that every complete N-vertex geometric graph whose edges are colored with q colors, yields a monochromatic convex geometric graph on n vertices. For fixed @?>=2 and q>=4, our results determine the correct exponential tower growth rate for N as a function of n, similar to the usual hypergraph Ramsey problem, even though we require our monochromatic set to be in convex position. Our results also apply to the case of @?=3 and q=2 by using a geometric variation of the stepping up lemma of Erdos and Hajnal. This is in contrast to the fact that the upper and lower bounds for the usual 3-uniform hypergraph Ramsey problem for two colors differ by one exponential in the tower.

For natural numbers d and t there exists a positive C such that if F is a family of n[superscript C] semi-algebraic sets in R[superscript d] of description complexity at most t, then there is a subset F' of F of size $n$ such that either every pair of elements in F' intersect or the elements of F' are pairwise disjoint. This result, which also holds if the intersection relation is replaced by any semi-algebraic relation of bounded description complexity, was proved by Alon, Pach, Pinchasi, Radoicic, and Sharir and improves on a bound of 4[superscript n] for the family F which follows from a straightforward application of Ramsey's theorem. We extend this semi-algebraic version of Ramsey's theorem to k-ary relations and give matching upper and lower bounds for the corresponding Ramsey function, showing that it grows as a tower of height k-1. This improves on a direct application of Ramsey's theorem by one exponential. We apply this result 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., Simons Foundation (Fellowship), National Science Foundation (U.S.) (Grant DMS 1069197), NEC Corporation (MIT Award), National Science Foundation (U.S.) (Postdoctoral Fellowship), Swiss National Science Foundation (Grant 200021-125287/1)

A graph drawn in the plane is called k-quasi-planar if it does not contain k pairwise crossing edges. It has been conjectured for a long time that for every fixed k, the maximum number of edges of a k-quasi-planar graph with n vertices is O(n). The best known upper bound is n(\log n)^{O(\log k)}. In the present note, we improve this bound to (n\log n)2^{\alpha^{c_k}(n)} in the special case where the graph is drawn in such a way that every pair of edges meet at most once. Here \alpha(n) denotes the (extremely slowly growing) inverse of the Ackermann function. We also make further progress on the conjecture for k-quasi-planar graphs in which every edge is drawn as an x-monotone curve. Extending some ideas of Valtr, we prove that the maximum number of edges of such graphs is at most 2^{ck^6}n\log n., Comment: arXiv admin note: substantial text overlap with arXiv:1106.0958

A curve $\gamma$ in the plane is $t$-monotone if its interior has at most $t-1$ vertical tangent points. A family of $t$-monotone curves $F$ is simple if any two members intersect at most once. It is shown that if $F$ is a simple family of $n$ $t$-monotone curves with at least $\epsilon n^2$ intersecting pairs (disjoint pairs), then there exists two subfamilies $F_1,F_2\subset F$ of size $\delta n$ each, such that every curve in $F_1$ intersects (is disjoint to) every curve in $F_2$, where $\delta$ depends only on $\epsilon$. We apply these results to find pairwise disjoint edges in simple topological graphs with $t$-monotone edges.

Let E S ( n ) ES(n) be the smallest integer such that any set of E S ( n ) ES(n) points in the plane in general position contains n n points in convex position. In their seminal 1935 paper, Erdős and Szekeres showed that E S ( n ) ≤ ( 2 n − 4 n − 2 ) + 1 = 4 n − o ( n ) ES(n) \leq {2n - 4\choose n-2} + 1 = 4^{n -o(n)} . In 1960, they showed that E S ( n ) ≥ 2 n − 2 + 1 ES(n) \geq 2^{n-2} + 1 and conjectured this to be optimal. In this paper, we nearly settle the Erdős-Szekeres conjecture by showing that E S ( n ) = 2 n + o ( n ) ES(n) =2^{n +o(n)} .

Let C be a family of n convex bodies in the plane, which can be decomposed into k subfamilies of pairwise disjoint sets. It is shown that the number of tangencies between the members of C is at most O(kn), and that this bound cannot be improved. If we only assume that our sets are connected and vertically convex, that is, their intersection with any vertical line is either a segment or the empty set, then the number of tangencies can be superlinear in n, but it cannot exceed O(n log(2) n). Our results imply a new upper bound on the number of regular intersection points on the boundary of boolean OR C. Published by Elsevier B.V.

We prove a Hadwiger transversal-type result, characterizing convex position on a family of non-crossing convex bodies in the plane. This theorem suggests a definition for the order type of a family of convex bodies, generalizing the usual definition of order type for point sets. This order type turns out to be an oriented matroid. We also give new upper bounds on the Erdős–Szekeres theorem in the context of convex bodies.

Let r(n) denote the largest integer such that every family \(\mathcal{C}\) of n pairwise disjoint segments in the plane in general position has r(n) members whose order type can be represented by points. Pach and Toth gave a construction that shows r(n) log16n. In this note, we will show that r(n) > cn1/4 for some absolute constant c.

We answer some of the questions raised by Golumbic, Lipshteyn and Stern and obtain some other results about edge intersection graphs of paths on a grid (EPG graphs). We show that for any d>=4, in order to represent every n vertex graph with maximum degree d as an edge intersection graph of n paths on a grid, a grid of area @Q(n^2) is needed. We also show several results related to the classes B"k-EPG, where B"k-EPG denotes the class of graphs that have an EPG representation such that each path has at most k bends. In particular, we prove: For a fixed k and a sufficiently large n, the complete bipartite graph K"m","n does not belong to B"2"m"-"3-EPG (it is known that this graph belongs to B"2"m"-"2-EPG); for any odd integer k we have B"k-EPG @?B"k"+"1-EPG; there is no number k such that all graphs belong to B"k-EPG; only 2^O^(^k^n^l^o^g^(^k^n^)^) out of all the 2^n^2 labeled graphs with n vertices are in B"k-EPG.

We provide several constructions for problems in Ramsey theory. First, we prove a superexponential lower bound for the classical 4-uniform Ramsey number $r_4(5,n)$, and the same for the iterated $(k-4)$-fold logarithm of the $k$-uniform version $r_k(k+1,n)$. This is the first improvement of the original exponential lower bound for $r_4(5,n)$ implicit in work of Erd\H os and Hajnal from 1972 and also improves the current best known bounds for larger $k$ due to the authors. Second, we prove an upper bound for the hypergraph Erd\H os-Rogers function $f^k_{k+1, k+2}(N)$ that is an iterated $(k-13)$-fold logarithm in $N$. This improves the previous upper bounds that were only logarithmic and addresses a question of Dudek and the first author that was reiterated by Conlon, Fox and Sudakov. Third, we generalize the results of Erd\H os and Hajnal about the 3-uniform Ramsey number of $K_4$ minus an edge versus a clique to $k$-uniform hypergraphs., Comment: arXiv admin note: text overlap with arXiv:1505.05767

The Ramsey number $r_k(s,n)$ is the minimum $N$ such that every red-blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ contains a red set of size $s$ or a blue set of size $n$, where a set is red (blue) if all of its $k$-subsets are red (blue). A $k$-uniform \emph{tight path} of size $s$, denoted by $P_{s}$, is a set of $s$ vertices $v_1 < \cdots < v_{s}$ in $\mathbb{Z}$, and all $s-k+1$ edges of the form $\{v_j,v_{j+1},\ldots, v_{j + k -1}\}$. Let $r_k(P_s, n)$ be the minimum $N$ such that every red-blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ results in a red $P_{s}$ or a blue set of size $n$. The problem of estimating both $r_k(s,n)$ and $r_k(P_s, n)$ for $k=2$ goes back to the seminal work of Erdos and Szekeres from 1935, while the case $k\ge 3$ was first investigated by Erdos and Rado in 1952. In this paper, we deduce a quantitative relationship between multicolor variants of $r_k(P_s, n)$ and $r_k(n, n)$. This yields several consequences including the following: (1) We determine the correct tower growth rate for both $r_k(s,n)$ and $r_k(P_s, n)$ for $s \ge k+3$. The question of determining the tower growth rate of $r_k(s,n)$ for all $s \ge k+1$ was posed by Erdos and Hajnal in 1972. (2) We show that determining the tower growth rate of $r_k(P_{k+1}, n)$ is equivalent to determining the tower growth rate of $r_k(n,n)$, which is a notorious conjecture of Erdos, Hajnal and Rado from 1965 that remains open. Some related off-diagonal hypergraph Ramsey problems are also explored.

Given a finite point set $P \subset \mathbb{R}^d$, a $k$-ary semi-algebraic relation $E$ on $P$ is the set of $k$-tuples of points in $P$, 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 $t$. The Ramsey number $R^{d,t}_k(s,n)$ is the minimum $N$ such that any $N$-element point set $P$ in $\mathbb{R}^d$ equipped with a $k$-ary semi-algebraic relation $E$, such that $E$ has complexity at most $t$, contains $s$ members such that every $k$-tuple induced by them is in $E$, or $n$ members such that every $k$-tuple induced by them is not in $E$. We give a new upper bound for $R^{d,t}_k(s,n)$ for $k\geq 3$ and $s$ fixed. In particular, we show that for fixed integers $d,t,s$, $R^{d,t}_3(s,n) \leq 2^{n^{o(1)}},$ establishing a subexponential upper bound on $R^{d,t}_3(s,n)$. This improves the previous bound of $2^{n^C}$ due to Conlon, Fox, Pach, Sudakov, and Suk, where $C$ is a very large constant depending on $d,t,$ and $s$. As an application, we give new estimates for a recently studied Ramsey-type problem on hyperplane arrangements in $\mathbb{R}^d$. We also study multi-color Ramsey numbers for triangles in our semi-algebraic setting, achieving some partial results.

It is shown that fora constant t is an element of N, every simple topological graph on n vertices has 0(n) edges if the graph has no two sets of t edges such that every edge in one set is disjoint from all edges of the other set (i.e., the complement of the intersection graph of the edges is K-t,K-t-free). As an application, we settle the tangled-thrackle conjecture formulated by Pach, Radoicic, and Toth: Every n-vertex graph drawn in the plane such that every pair of edges have precisely one point in common, where this point is either a common endpoint, a crossing, or a point of tangency, has at most O(n) edges. (C) 2015 Elsevier Ltd. All rights reserved.

We prove that in a simple matroid, the maximal number of joints formed by L lines is o(L[superscript 2]) and Ω(L[superscript 2-ε]) for any ε > 0. Keywords: Matroids, The joints problem, Arithmetic progression, National Science Foundation (U.S.) (Postdoctoral Fellowship)

A curve γ in the plane is t-monotone if its interior has at most t−1 vertical tangent points. A family of t-monotone curves F is simple if any two members intersect at most once. It is shown that if F is a simple family of nt-monotone curves with at least en2 intersecting pairs (disjoint pairs), then there exists two subfamilies F1,F2⊂F of size δn each, such that every curve in F1 intersects (is disjoint to) every curve in F2, where δ depends only on e. We apply these results to find pairwise disjoint edges in simple topological graphs.

Let ni¾?li¾?2 and qi¾?2. We consider the minimum N such that whenever we have N points in the plane in general position and the l-subsets of these points are colored with q colors, there is a subset S of n points all of whose l-subsets have the same color and furthermore S is in convex position. This combines two classical areas of intense study over the last 75 years: the Ramsey problem for hypergraphs and the Erdi¾?s-Szekeres theorem on convex configurations in the plane. For the special case l=2, we establish a single exponential bound on the minimum N such that every complete N-vertex geometric graph whose edges are colored with q colors, yields a monochromatic convex geometric graph on n vertices. For fixed li¾?2 and qi¾?4, our results determine the correct exponential tower growth rate for N as a function of n, similar to the usual hypergraph Ramsey problem, even though we require our monochromatic set to be in convex position. Our results also apply to the case of l=3 and q=2 by using a geometric variation of the Stepping-up lemma of Erdi¾?s and Hajnal. This is in contrast to the fact that the upper and lower bounds for the usual 3-uniform hypergraph Ramsey problem for two colors differ by one exponential in the tower.

In this note, we prove several Turan-type results on geometric hypergraphs. The two main theorems are (1) every n-vertex geometric 3-hypergraph in the plane with no three strongly crossing edges has at most O(n 2) edges, and (2) every n-vertex geometric 3-hypergraph in 3-space with no two disjoint edges has at most O(n 2) edges. These results support two conjectures that were raised by Dey and Pach, and by Akiyama and Alon.

A geometric graph is a graph drawn in the plane with vertices represented by points and edges as straight-line segments. A geometric graph contains a (k,l)-crossing family if there is a pair of edge subsets E 1, E 2 such that | E 1 | = k and | E 2 | = l, the edges in E 1 are pairwise crossing, the edges in E 2 are pairwise crossing, and every edge in E 1 is disjoint to every edge in E 2. We conjecture that for any fixed k, l, every n-vertex geometric graph with no (k, l)-crossing family has at most c k, l n edges, where c k, l is a constant that depends only on k and l. In this note, we show that every n-vertex geometric graph with no (k, k)-crossing family has at most c k nlogn edges, where c k is a constant that depends only on k, by proving a more general result that relates an extremal function of a geometric graph F with an extremal function of two completely disjoint copies of F. We also settle the conjecture for geometric graphs with no (2, 1)-crossing family. As a direct application, this implies that for any circle graph F on three vertices, every n-vertex geometric graph that does not contain a matching whose intersection graph is F has at most O(n) edges.

A topological graph is k-quasi-planar if it does not contain k pairwise crossing edges. A topological graph is simple if every pair of its edges intersect at most once (either at a vertex or at their intersection). In 1996, Pach, Shahrokhi, and Szegedy [16] showed that every n-vertex simple k-quasi-planar graph contains at most $O\left(n(\log n)^{2k-4}\right)$ edges. This upper bound was recently improved (for large k) by Fox and Pach [8] to n(logn)O(logk). In this note, we show that all such graphs contain at most $(n\log^2n )2^{\alpha^{c_k}(n)}$ edges, where α(n) denotes the inverse Ackermann function and ck is a constant that depends only on k.

Dedicated to the 75th anniversary of the publication of the Happy Ending Theorem, 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\ge k\ge 2 and q\ge 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 Erd\H os and Szekeres that N_2(q,n)=(n-1)^q+1 and N_3(2,n) = {2n -4\choose n-2} + 1. Determining the other values of these functions appears to be a difficult task. Here we show that 2^{(n/q)^{q-1}} \leq N_3(q,n) \leq 2^{n^{q-1}\log n}, for q \geq 2 and n \geq q+2. Using a stepping-up approach that goes back to Erdos 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., National Science Foundation (U.S.) (NSF-CAREER Award (DMS-0812005), Massachusetts Institute of Technology (Simons Fellowship), United States-Israel Binational Science Foundation (grant)

It is shown that every complete $$n$$ -vertex simple topological graph has at $$\varOmega (n^{1/3})$$ pairwise disjoint edges, and these edges can be found in polynomial time. This proves a conjecture of Pach and Toth, which appears as Problem 5 from Chapter 9.5 in Research Problems in Discrete Geometry by Brass, Moser, and Pach.

We give a short proof that for any fixed integer $k$, the maximum number size of a $K_{k,k}$-cross free family is linear in the size of the groundset. We also give tight bounds on the maximum size of a $K_k$-cross free family in the case when ${\cal F}$ is intersecting or an antichain.

A topological graph G is a graph drawn in the plane with vertices represented by points and edges represented by continuous arcs connecting the vertices. If every edge is drawn as a straight-line segment, then G is called a geometric graph. A k-grid in a topological graph is a pair of subsets of the edge set, each of size k, such that every edge in one subset crosses every edge in the other subset. It is known that every n-vertex topological graph with no k-grid has O-k(n) edges. We conjecture that the number of edges of every n-vertex topological graph with no k-grid such that all of its 2k edges have distinct endpoints is 0 k(n). This conjecture is shown to be true apart from an iterated logarithmic factor log* n. A k-grid is natural if its edges have distinct endpoints, and the arcs representing each of its edge subsets are pairwise disjoint. We also conjecture that every n-vertex geometric graph with no natural k-grid has Ok(n) edges, but we can establish only an O-k(n log(2) n) upper bound. We verify the above conjectures in several special cases. (C) 2014 Elsevier B.V. All rights reserved.

A straight-line drawing of a graph $G$ is a mapping which assigns to each vertex a point in the plane and to each edge a straight-line segment connecting the corresponding two points. The rectilinear crossing number of a graph $G$, $\overline{cr}(G)$, is the minimum number of crossing edges in any straight-line drawing of $G$. Determining or estimating $\overline{cr}(G)$ appears to be a difficult problem, and deciding if $\overline{cr}(G)\leq k$ is known to be NP-hard. In fact, the asymptotic behavior of $\overline{cr}(K_n)$ is still unknown. In this paper, we present a deterministic $n^{2+o(1)}$-time algorithm that finds a straight-line drawing of any $n$-vertex graph $G$ with $\overline{cr}(G) + o(n^4)$ crossing edges. Together with the well-known Crossing Lemma due to Ajtai et al. and Leighton, this result implies that for any dense $n$-vertex graph $G$, one can efficiently find a straight-line drawing of $G$ with $(1 + o(1))\overline{cr}(G)$ crossing edges., Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016)

A bipartite graph $G$ is semi-algebraic in $\mathbb{R}^d$ if its vertices are represented by point sets $P,Q \subset \mathbb{R}^d$ and its edges are defined as pairs of points $(p,q) \in P\times Q$ 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 $K_{k,k}$-free semi-algebraic bipartite graph $G = (P,Q,E)$ in $\mathbb{R}^2$ with $|P| = m$ and $|Q| = n$ is at most $O((mn)^{2/3} + m + n)$, and this bound is tight. In dimensions $d \geq 3$, we show that all such semi-algebraic graphs have at most $C\left((mn)^{ \frac{d}{d+1} + \varepsilon} + m + n\right)$ edges, where here $\varepsilon$ is an arbitrarily small constant and $C = C(d,k,t,\varepsilon)$. This result is a far-reaching generalization of the classical Szemer\'edi-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 $\mathbb{R}^d$, an improved bound for a $d$-dimensional variant of the Erd\H{o}s unit distances problem, and more.

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 delta > 0 such that the vertex classes P-1,...,P-k of every k -partite k -uniform semialgebraic hypergraph H = (P-1 U center dot center dot center dot U P-k, E) with vertical bar E vertical bar >= epsilon ll(j=1)(k) vertical bar P-i vertical bar have, for 1 d) have, for 1 0 the vertex set can be equitably partitioned into a bounded number of parts (in terms of epsilon and the complexity) so that all but an epsilon-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/epsilon. 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.

A semi-algebraic graph $G = (V,E)$ is a graph where the vertices are points in $\mathbb{R}^d$, and the edge set $E$ is defined by a semi-algebraic relation of constant complexity on $V$. In this note, we establish the following Ramsey-Turán theorem: for every integer $p\geq 3$, every $K_{p}$-free semi-algebraic graph on $n$ vertices with independence number $o(n)$ has at most $\frac{1}{2}\left(1 - \frac{1}{\lceil p/2\rceil-1} + o(1) \right)n^2$ edges. Here, the dependence on the complexity of the semi-algebraic relation is hidden in the $o(1)$ term. Moreover, we show that this bound is tight.