arXiv subjects

Janos Pach

Publications and source records attributed to Janos Pach.

36 records · Page 2Linked to original sources

arXiv2019

Coloring Hasse diagrams and disjointness graphs of curves

Given a family of curves $\mathcal{C}$ in the plane, its disjointness graph is the graph whose vertices correspond to the elements of $\mathcal{C}$, and two vertices are joined by an edge if and only if the corresponding sets are disjoint. We prove that for every positive integer $r$ and $n$, there exists a family of $n$ curves whose disjointness graph has girth $r$ and chromatic number $Ω(\frac{1}{r}\log n)$. In the process we slightly improve Bollobás's old result on Hasse diagrams and show that our improved bound is best possible for uniquely generated partial orders.

math.CO↗

arXiv2019

Ordered graphs and large bi-cliques in intersection graphs of curves

An ordered graph $G_<$ is a graph with a total ordering $<$ on its vertex set. A monotone path of length $k$ is a sequence of vertices $v_1 0$ such that every ordered graph on $n$ vertices that does not contain a monotone path of length $k$ as an induced subgraph has a vertex of degree at least $c_kn$, or its complement has a bi-clique of size at least $c_kn/\log n$. A similar result holds for ordered graphs containing no induced ordered subgraph isomorphic to a fixed ordered matching. As a consequence, we give a short combinatorial proof of the following theorem of Fox and Pach. There exists a constant $c>0$ such the intersection graph $G$ of any collection of $n$ $x$-monotone curves in the plane has a bi-clique of size at least $cn/\log n$ or its complement contains a bi-clique of size at least $cn$. (A curve is called $x$-monotone if every vertical line intersects it in at most one point.) We also prove that if $G$ has at most $\left(\frac14 -ε\right){n\choose 2}$ edges for some $ε>0$, then $\overline{G}$ contains a linear sized bi-clique. We show that this statement does not remain true if we replace $\frac14$ by any larger constants.

math.CO↗

arXiv2018

Semi-algebraic colorings of complete graphs

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 $p\ge 3$ and $m\ge 2$, the classical Ramsey number $R(p;m)$ is the smallest positive integer $n$ such that any $m$-coloring of the edges of $K_n$, the complete graph on $n$ vertices, contains a monochromatic $K_p$. It is a longstanding open problem that goes back to Schur (1916) to decide whether $R(p;m)=2^{O(m)}$, for a fixed $p$. We prove that this is true if each color class is defined semi-algebraically with bounded complexity. The order of magnitude of this bound is tight. Our proof is based on the Cutting Lemma of Chazelle {\em et al.}, and on a Szemerédi-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.

math.CO↗

arXiv2018

On the chromatic number of disjointness graphs of curves

Let $ω(G)$ and $χ(G)$ denote the clique number and chromatic number of a graph $G$, respectively. The {\em disjointness graph} of a family of curves (continuous arcs in the plane) is the graph whose vertices correspond to the curves and in which two vertices are joined by an edge if and only if the corresponding curves are disjoint. A curve is called {\em $x$-monotone} if every vertical line intersects it in at most one point. An $x$-monotone curve is {\em grounded} if its left endpoint lies on the $y$-axis. We prove that if $G$ is the disjointness graph of a family of grounded $x$-monotone curves such that $ω(G)=k$, then $χ(G)\leq \binom{k+1}{2}$. If we only require that every curve is $x$-monotone and intersects the $y$-axis, then we have $χ(G)\leq \frac{k+1}{2}\binom{k+2}{3}$. Both of these bounds are best possible. The construction showing the tightness of the last result settles a 25 years old problem: it yields that there exist $K_k$-free disjointness graphs of $x$-monotone curves such that any proper coloring of them uses at least $Ω(k^{4})$ colors. This matches the upper bound up to a constant factor.

math.CO↗

arXiv2018

A stability theorem on cube tessellations

It is shown that if a $d$-dimensional cube is decomposed into n cubes, the side lengths of which belong to the interval $\left(1-\frac{1}{n^{1/d}+1}, 1\right], then $n$ is a perfect $d$-th power and all cubes are of the same size. This result is essentially tight.

math.CO↗

arXiv2018

Tiling the plane with equilateral triangles

Let $\cal T$ be a tiling of the plane with equilateral triangles no two of which share a side. We prove that if the side lengths of the triangles are bounded from below by a positive constant, then $\cal T$ is periodic and it consists of translates of only at most three different triangles. As a corollary, we prove a theorem of Scherer and answer a question of Nandakumar. The same result has been obtained independently by Richter and Wirth.

math.CO↗

arXiv2018

Controlling Lipschitz functions

Given any positive integers $m$ and $d$, we say the a sequence of points $(x_i)_{i\in I}$ in $\mathbb R^m$ is {\em Lipschitz-$d$-controlling} if one can select suitable values $y_i\; (i\in I)$ such that for every Lipschitz function $f:\mathbb R^m\rightarrow \mathbb R^d$ there exists $i$ with $|f(x_i)-y_i|<1$. We conjecture that for every $m\le d$, a sequence $(x_i)_{i\in I}\subset\mathbb R^m$ is $d$-controlling if and only if $$\sup_{n\in\mathbb N}\frac{|\{i\in I\, :\, |x_i|\le n\}|}{n^d}=\infty.$$ We prove that this condition is necessary and a slightly stronger one is already sufficient for the sequence to be $d$-controlling. We also prove the conjecture for $m=1$.

math.FA↗

arXiv2018

A crossing lemma for multigraphs

Let $G$ be a drawing of a graph with $n$ vertices and $e>4n$ edges, in which no two adjacent edges cross and any pair of independent edges cross at most once. According to the celebrated Crossing Lemma of Ajtai, Chvátal, Newborn, Szemerédi and Leighton, the number of crossings in $G$ is at least $c{e^3\over n^2}$, for a suitable constant $c>0$. In a seminal paper, Székely generalized this result to multigraphs, establishing the lower bound $c{e^3\over mn^2}$, where $m$ denotes the maximum multiplicity of an edge in $G$. We get rid of the dependence on $m$ by showing that, as in the original Crossing Lemma, the number of crossings is at least $c'{e^3\over n^2}$ for some $c'>0$, provided that the "lens" enclosed by every pair of parallel edges in $G$ contains at least one vertex. This settles a conjecture of Kaufmann.

math.CO↗

arXiv2017

Many Touchings Force Many Crossings

Given $n$ continuous open curves in the plane, we say that a pair is touching if they have only one interior point in common and at this point the first curve does not get from one side of the second curve to its other side. Otherwise, if the two curves intersect, they are said to form a crossing pair. Let $t$ and $c$ denote the number of touching pairs and crossing pairs, respectively. We prove that $c \ge {1\over 10^5}{t^2\over n^2}$, provided that $t\ge 10n$. Apart from the values of the constants, this result is best possible.

math.CO↗

arXiv2016

A polynomial regularity lemma for semi-algebraic hypergraphs and its applications in geometry and property testing

Fox, Gromov, Lafforgue, Naor, and Pach proved 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. 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 semi-algebraic hypergraphs of bounded complexity.

math.CO↗

arXiv2016

Approximating the rectilinear crossing number

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.

cs.CG↗

arXiv2015

Decomposition of a cube into nearly equal smaller cubes

Let $d$ be a fixed positive integer and let $ε>0$. It is shown that for every sufficiently large $n\geq n_0(d,ε)$, the $d$-dimensional unit cube can be decomposed into exactly $n$ smaller cubes such that the ratio of the side length of the largest cube to the side length of the smallest one is at most $1+ε$. Moreover, for every $n\geq n_0$, there is a decomposition with the required properties, using cubes of at most $d+2$ different side lengths. If we drop the condition that the side lengths of the cubes must be roughly equal, it is sufficient to use cubes of two different sizes.

math.CO↗

arXiv2013

Applications of a new separator theorem for string graphs

An intersection graph of curves in the plane is called a string graph. Matousek almost completely settled a conjecture of the authors by showing that every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log m). In the present note, this bound is combined with a result of the authors, according to which every dense string graph contains a large complete balanced bipartite graph. Three applications are given concerning string graphs G with n vertices: (i) if K_t is not a subgraph of G for some t, then the chromatic number of G is at most (\log n)^{O(\log t)}; (ii) if K_{t,t} is not a subgraph of G, then G has at most t(\log t)^{O(1)}n edges,; and (iii) a lopsided Ramsey-type result, which shows that the Erdos-Hajnal conjecture almost holds for string graphs.

math.CO↗

arXiv2011

The number of edges in k-quasi-planar graphs

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^{α^{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 α(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.

math.CO↗

arXiv2011

Erdos-Szekeres-type theorems for monotone paths and convex bodies

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.

math.CO↗

arXiv2011

Monotone drawings of planar graphs

Let G be a graph drawn in the plane so that its edges are represented by x-monotone curves, any pair of which cross an even number of times. We show that G can be redrawn in such a way that the x-coordinates of the vertices remain unchanged and the edges become non-crossing straight-line segments.

math.CO↗

arXiv2010

Overlap properties of geometric expanders

The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\in (0,1]$ such that no matter how we map the vertices of $H$ into $\R^d$, there is a point covered by at least a $c(H)$-fraction of the simplices induced by the images of its hyperedges. In~\cite{Gro2}, motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, it was asked whether or not there exists a sequence $\{H_n\}_{n=1}^\infty$ of arbitrarily large $(d+1)$-uniform hypergraphs with bounded degree, for which $\inf_{n\ge 1} c(H_n)>0$. Using both random methods and explicit constructions, we answer this question positively by constructing infinite families of $(d+1)$-uniform hypergraphs with bounded degree such that their overlap numbers are bounded from below by a positive constant $c=c(d)$. We also show that, for every $d$, the best value of the constant $c=c(d)$ that can be achieved by such a construction is asymptotically equal to the limit of the overlap numbers of the complete $(d+1)$-uniform hypergraphs with $n$ vertices, as $n\rightarrow\infty$. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any $d$ and any $ε>0$, there exists $K=K(ε,d)\ge d+1$ satisfying the following condition. For any $k\ge K$, for any point $q \in \mathbb{R}^d$ and for any finite Borel measure $μ$ on $\mathbb{R}^d$ with respect to which every hyperplane has measure $0$, there is a partition $\mathbb{R}^d=A_1 \cup \ldots \cup A_{k}$ into $k$ measurable parts of equal measure such that all but at most an $ε$-fraction of the $(d+1)$-tuples $A_{i_1},\ldots,A_{i_{d+1}}$ have the property that either all simplices with one vertex in each $A_{i_j}$ contain $q$ or none of these simplices contain $q$.

math.CO↗

arXiv2010

A computational approach to Conway's thrackle conjecture

A drawing of a graph in the plane is called a thrackle if every pair of edges meets precisely once, either at a common vertex or at a proper crossing. Let t(n) denote the maximum number of edges that a thrackle of n vertices can have. According to a 40 years old conjecture of Conway, t(n)=n for every n>2. For any eps>0, we give an algorithm terminating in e^{O((1/eps^2)ln(1/eps))} steps to decide whether t(n)<(1+eps)n for all n>2. Using this approach, we improve the best known upper bound, t(n)<=3/2(n-1), due to Cairns and Nikolayevsky, to 167/117n<1.428n.

math.CO↗