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**Published in:** LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

A curve in the plane is x-monotone if every vertical line intersects it at most once. A family of curves are called pseudo-segments if every pair of them have at most one point in common. We construct 2^Ω(n^{4/3}) families, each consisting of n labelled x-monotone pseudo-segments such that their intersection graphs are different. On the other hand, we show that the number of such intersection graphs is at most 2^O(n^{3/2-ε}), where ε > 0 is a suitable constant. Our proof uses an upper bound on the number of set systems of size m on a ground set of size n, with VC-dimension at most d. Much better upper bounds are obtained if we only count bipartite intersection graphs, or, in general, intersection graphs with bounded chromatic number.

Jacob Fox, János Pach, and Andrew Suk. Enumeration of Intersection Graphs of x-Monotone Curves. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 4:1-4:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{fox_et_al:LIPIcs.GD.2024.4, author = {Fox, Jacob and Pach, J\'{a}nos and Suk, Andrew}, title = {{Enumeration of Intersection Graphs of x-Monotone Curves}}, booktitle = {32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)}, pages = {4:1--4:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-343-0}, ISSN = {1868-8969}, year = {2024}, volume = {320}, editor = {Felsner, Stefan and Klein, Karsten}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.4}, URN = {urn:nbn:de:0030-drops-212887}, doi = {10.4230/LIPIcs.GD.2024.4}, annote = {Keywords: Enumeration, intersection graphs, pseudo-segments, x-monotone} }

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**Published in:** LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

A complete geometric graph consists of a set P of n points in the plane, in general position, and all segments (edges) connecting them. It is a well known question of Bose, Hurtado, Rivera-Campo, and Wood, whether there exists a positive constant c < 1, such that every complete geometric graph on n points can be partitioned into at most cn plane graphs (that is, noncrossing subgraphs). We answer this question in the affirmative in the special case where the underlying point set P is dense, which means that the ratio between the maximum and the minimum distances in P is of the order of Θ(√n).

Adrian Dumitrescu and János Pach. Partitioning Complete Geometric Graphs on Dense Point Sets into Plane Subgraphs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 9:1-9:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dumitrescu_et_al:LIPIcs.GD.2024.9, author = {Dumitrescu, Adrian and Pach, J\'{a}nos}, title = {{Partitioning Complete Geometric Graphs on Dense Point Sets into Plane Subgraphs}}, booktitle = {32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)}, pages = {9:1--9:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-343-0}, ISSN = {1868-8969}, year = {2024}, volume = {320}, editor = {Felsner, Stefan and Klein, Karsten}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.9}, URN = {urn:nbn:de:0030-drops-212939}, doi = {10.4230/LIPIcs.GD.2024.9}, annote = {Keywords: Convexity, complete geometric Graph, crossing Family, plane Subgraph} }

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**Published in:** Dagstuhl Reports, Volume 14, Issue 2 (2024)

This report documents the program and the outcomes of Dagstuhl Seminar 24062 "Beyond-Planar Graphs: Models, Structures and Geometric Representations". The seminar investigated beyond-planar graphs, in particular, their combinatorial and topological structures, computational complexity and algorithmics for recognition, geometric representations, and their applications to real-world network visualization. Compared to the previous two editions of the seminar, we focus more on aspects of combinatorics and geometry.
The program consists of four invited talks on beyond planar graphs, open problem session, problem solving sessions and progress report sessions. Specific open problems include questions regarding the combinatorial structures and topology (e.g., k^+-real face graphs, beyond upward planar graphs, sparse universal geometric graphs, local-crossing-critical graphs), the geometric representations (e.g., constrained outer string graphs, rerouting curves on surface), and applications.
The details of the invited talks and progress reports from each working groups are included in this report.

Vida Dujmović, Seok-Hee Hong, Michael Kaufmann, János Pach, and Henry Förster. Beyond-Planar Graphs: Models, Structures and Geometric Representations (Dagstuhl Seminar 24062). In Dagstuhl Reports, Volume 14, Issue 2, pp. 71-94, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@Article{dujmovic_et_al:DagRep.14.2.71, author = {Dujmovi\'{c}, Vida and Hong, Seok-Hee and Kaufmann, Michael and Pach, J\'{a}nos and F\"{o}rster, Henry}, title = {{Beyond-Planar Graphs: Models, Structures and Geometric Representations (Dagstuhl Seminar 24062)}}, pages = {71--94}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2024}, volume = {14}, number = {2}, editor = {Dujmovi\'{c}, Vida and Hong, Seok-Hee and Kaufmann, Michael and Pach, J\'{a}nos and F\"{o}rster, Henry}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.14.2.71}, URN = {urn:nbn:de:0030-drops-204999}, doi = {10.4230/DagRep.14.2.71}, annote = {Keywords: Combinatorial geometry, Graph algorithm, Graph drawing, Graph theory, Network visualization} }

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**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in ℝ^d and any radius satisfies β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions, which prove that this upper bound is asymptotically tight. For example, we describe a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of the Čech complex at the other radius is n². In particular, there is an arrangement of n contruent balls in ℝ³ that enclose a quadratic number of voids, which answers a long-standing open question in computational geometry.

Herbert Edelsbrunner and János Pach. Maximum Betti Numbers of Čech Complexes. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 53:1-53:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{edelsbrunner_et_al:LIPIcs.SoCG.2024.53, author = {Edelsbrunner, Herbert and Pach, J\'{a}nos}, title = {{Maximum Betti Numbers of \v{C}ech Complexes}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {53:1--53:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.53}, URN = {urn:nbn:de:0030-drops-199981}, doi = {10.4230/LIPIcs.SoCG.2024.53}, annote = {Keywords: Discrete geometry, computational topology, \v{C}ech complexes, Delaunay mosaics, Alpha complexes, Betti numbers, extremal questions} }

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**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

We prove a far-reaching strengthening of Szemerédi’s regularity lemma for intersection graphs of pseudo-segments. It shows that the vertex set of such graphs can be partitioned into a bounded number of parts of roughly the same size such that almost all of the bipartite graphs between pairs of parts are complete or empty. We use this to get an improved bound on disjoint edges in simple topological graphs, showing that every n-vertex simple topological graph with no k pairwise disjoint edges has at most n(log n)^O(log k) edges.

Jacob Fox, János Pach, and Andrew Suk. A Structure Theorem for Pseudo-Segments and Its Applications. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 59:1-59:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{fox_et_al:LIPIcs.SoCG.2024.59, author = {Fox, Jacob and Pach, J\'{a}nos and Suk, Andrew}, title = {{A Structure Theorem for Pseudo-Segments and Its Applications}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {59:1--59:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.59}, URN = {urn:nbn:de:0030-drops-200040}, doi = {10.4230/LIPIcs.SoCG.2024.59}, annote = {Keywords: Regularity lemma, pseudo-segments, intersection graphs} }

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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

The disjointness graph of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph G of any system of segments in the plane is χ-bounded, that is, its chromatic number χ(G) is upper bounded by a function of its clique number ω(G).
Here we show that this statement does not remain true for systems of polygonal chains of length 2. We also construct systems of polygonal chains of length 3 such that their disjointness graphs have arbitrarily large girth and chromatic number. In the opposite direction, we show that the class of disjointness graphs of (possibly self-intersecting) 2-way infinite polygonal chains of length 3 is χ-bounded: for every such graph G, we have χ(G) ≤ (ω(G))³+ω(G).

János Pach, Gábor Tardos, and Géza Tóth. Disjointness Graphs of Short Polygonal Chains. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 56:1-56:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{pach_et_al:LIPIcs.SoCG.2022.56, author = {Pach, J\'{a}nos and Tardos, G\'{a}bor and T\'{o}th, G\'{e}za}, title = {{Disjointness Graphs of Short Polygonal Chains}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {56:1--56:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.56}, URN = {urn:nbn:de:0030-drops-160645}, doi = {10.4230/LIPIcs.SoCG.2022.56}, annote = {Keywords: chi-bounded, disjointness graph} }

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**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Given a family F of k-element sets, S₁,…,S_r ∈ F form an r-sunflower if S_i ∩ S_j = S_{i'} ∩ S_{j'} for all i ≠ j and i' ≠ j'. According to a famous conjecture of Erdős and Rado (1960), there is a constant c = c(r) such that if |F| ≥ c^k, then F contains an r-sunflower.
We come close to proving this conjecture for families of bounded Vapnik-Chervonenkis dimension, VC-dim(F) ≤ d. In this case, we show that r-sunflowers exist under the slightly stronger assumption |F| ≥ 2^{10k(dr)^{2log^{*} k}}. Here, log^* denotes the iterated logarithm function.
We also verify the Erdős-Rado conjecture for families F of bounded Littlestone dimension and for some geometrically defined set systems.

Jacob Fox, János Pach, and Andrew Suk. Sunflowers in Set Systems of Bounded Dimension. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 37:1-37:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{fox_et_al:LIPIcs.SoCG.2021.37, author = {Fox, Jacob and Pach, J\'{a}nos and Suk, Andrew}, title = {{Sunflowers in Set Systems of Bounded Dimension}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {37:1--37:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.37}, URN = {urn:nbn:de:0030-drops-138366}, doi = {10.4230/LIPIcs.SoCG.2021.37}, annote = {Keywords: Sunflower, VC-dimension, Littlestone dimension, pseudodisks} }

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**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

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.

Jacob Fox, János Pach, and Andrew Suk. Bounded VC-Dimension Implies the Schur-Erdős Conjecture. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 46:1-46:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fox_et_al:LIPIcs.SoCG.2020.46, author = {Fox, Jacob and Pach, J\'{a}nos and Suk, Andrew}, title = {{Bounded VC-Dimension Implies the Schur-Erd\H{o}s Conjecture}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {46:1--46:8}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.46}, URN = {urn:nbn:de:0030-drops-122046}, doi = {10.4230/LIPIcs.SoCG.2020.46}, annote = {Keywords: Ramsey theory, VC-dimension, Multicolor Ramsey numbers} }

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**Published in:** Dagstuhl Reports, Volume 9, Issue 2 (2019)

This report documents the program and the outcomes of Dagstuhl Seminar 19092 "Beyond-Planar Graphs: Combinatorics, Models and Algorithms" which brought together 36 researchers in the areas of graph theory, combinatorics, computational geometry, and graph
drawing. This seminar continued the work initiated in Dagstuhl Seminar 16452 "Beyond-Planar Graphs: Algorithmics and Combinatorics" and focused on the exploration of structural properties and the development of algorithms for so-called beyond-planar graphs, i.e., non-planar graphs that admit a drawing with topological constraints such as specific types of crossings, or with some forbidden crossing patterns.
The seminar began with four talks about the results of scientific collaborations originating from the previous Dagstuhl seminar. Next we discussed open research problems
about beyond planar graphs, such as their combinatorial structures (e.g., book thickness, queue number), their topology (e.g., simultaneous embeddability, gap planarity, quasi-quasiplanarity), their geometric representations
(e.g., representations on few segments or arcs), and applications
(e.g., manipulation of graph drawings by untangling operations). Six working groups were formed that investigated several of the open research questions. In addition, talks on related subjects and recent conference contributions were presented in the morning opening sessions. Abstracts of all talks and a report from each working group are included in this report.

Seok-Hee Hong, Michael Kaufmann, János Pach, and Csaba D. Tóth. Beyond-Planar Graphs: Combinatorics, Models and Algorithms (Dagstuhl Seminar 19092). In Dagstuhl Reports, Volume 9, Issue 2, pp. 123-156, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Article{hong_et_al:DagRep.9.2.123, author = {Hong, Seok-Hee and Kaufmann, Michael and Pach, J\'{a}nos and T\'{o}th, Csaba D.}, title = {{Beyond-Planar Graphs: Combinatorics, Models and Algorithms (Dagstuhl Seminar 19092)}}, pages = {123--156}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2019}, volume = {9}, number = {2}, editor = {Hong, Seok-Hee and Kaufmann, Michael and Pach, J\'{a}nos and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.9.2.123}, URN = {urn:nbn:de:0030-drops-108634}, doi = {10.4230/DagRep.9.2.123}, annote = {Keywords: combinatorial geometry, geometric algorithms, graph algorithms, graph drawing, graph theory, network visualization} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

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 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, 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 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.

Jacob Fox, János Pach, and Andrew Suk. Semi-Algebraic Colorings of Complete Graphs. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 36:1-36:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fox_et_al:LIPIcs.SoCG.2019.36, author = {Fox, Jacob and Pach, J\'{a}nos and Suk, Andrew}, title = {{Semi-Algebraic Colorings of Complete Graphs}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {36:1--36:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.36}, URN = {urn:nbn:de:0030-drops-104401}, doi = {10.4230/LIPIcs.SoCG.2019.36}, annote = {Keywords: Semi-algebraic graphs, Ramsey theory, regularity lemma} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Let omega(G) and chi(G) denote the clique number and chromatic number of a graph G, respectively. The 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 x-monotone if every vertical line intersects it in at most one point. An x-monotone curve is 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 omega(G)=k, then chi(G)<= binom{k+1}{2}. If we only require that every curve is x-monotone and intersects the y-axis, then we have chi(G)<= 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 Omega(k^{4}) colors. This matches the upper bound up to a constant factor.

János Pach and István Tomon. On the Chromatic Number of Disjointness Graphs of Curves. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 54:1-54:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{pach_et_al:LIPIcs.SoCG.2019.54, author = {Pach, J\'{a}nos and Tomon, Istv\'{a}n}, title = {{On the Chromatic Number of Disjointness Graphs of Curves}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {54:1--54:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.54}, URN = {urn:nbn:de:0030-drops-104582}, doi = {10.4230/LIPIcs.SoCG.2019.54}, annote = {Keywords: string graph, chromatic number, intersection graph} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

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 Bekos, Kaufmann, and Raftopoulou.

János Pach and Géza Tóth. A Crossing Lemma for Multigraphs. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 65:1-65:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{pach_et_al:LIPIcs.SoCG.2018.65, author = {Pach, J\'{a}nos and T\'{o}th, G\'{e}za}, title = {{A Crossing Lemma for Multigraphs}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {65:1--65:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.65}, URN = {urn:nbn:de:0030-drops-87781}, doi = {10.4230/LIPIcs.SoCG.2018.65}, annote = {Keywords: crossing number, Crossing Lemma, multigraph, separator theorem} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

A string graph is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on n vertices can be partitioned into five cliques such that some pair of them is not connected by any edge (n --> infty). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollary, we obtain that almost all string graphs on n vertices are intersection graphs of plane convex sets.

János Pach, Bruce Reed, and Yelena Yuditsky. Almost All String Graphs are Intersection Graphs of Plane Convex Sets. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 68:1-68:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{pach_et_al:LIPIcs.SoCG.2018.68, author = {Pach, J\'{a}nos and Reed, Bruce and Yuditsky, Yelena}, title = {{Almost All String Graphs are Intersection Graphs of Plane Convex Sets}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {68:1--68:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.68}, URN = {urn:nbn:de:0030-drops-87818}, doi = {10.4230/LIPIcs.SoCG.2018.68}, annote = {Keywords: String graph, intersection graph, plane convex set} }

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**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

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 Erdos 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 Erdos-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/epsilon)^{O(d)}, improving the original bound of (1/epsilon)^{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 in the k-uniform setting.

Jacob Fox, János Pach, and Andrew Suk. Erdös-Hajnal Conjecture for Graphs with Bounded VC-Dimension. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 43:1-43:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{fox_et_al:LIPIcs.SoCG.2017.43, author = {Fox, Jacob and Pach, J\'{a}nos and Suk, Andrew}, title = {{Erd\"{o}s-Hajnal Conjecture for Graphs with Bounded VC-Dimension}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {43:1--43:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.43}, URN = {urn:nbn:de:0030-drops-72246}, doi = {10.4230/LIPIcs.SoCG.2017.43}, annote = {Keywords: VC-dimension, Ramsey theory, regularity lemma} }

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**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

The disjointness graph G=G(S) of a set of segments S in R^d, d>1 is a graph whose vertex set is S and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of G satisfies chi(G)<=omega(G)^4+omega(G)^3 where omega(G) denotes the clique number of G. It follows, that S has at least cn^{1/5} pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments.
We show that computing omega(G) and chi(G) for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colorings of G in which the number of colors satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free (omega(G)=2), but whose chromatic numbers are arbitrarily large.

János Pach, Gábor Tardos, and Géza Tóth. Disjointness Graphs of Segments. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 59:1-59:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{pach_et_al:LIPIcs.SoCG.2017.59, author = {Pach, J\'{a}nos and Tardos, G\'{a}bor and T\'{o}th, G\'{e}za}, title = {{Disjointness Graphs of Segments}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {59:1--59:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.59}, URN = {urn:nbn:de:0030-drops-71960}, doi = {10.4230/LIPIcs.SoCG.2017.59}, annote = {Keywords: disjointness graph, chromatic number, clique number, chi-bounded} }

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**Published in:** Dagstuhl Reports, Volume 6, Issue 11 (2017)

This report summarizes Dagstuhl Seminar 16452 "Beyond-Planar Graphs: Algorithmics and Combinatorics'' and documents the talks and discussions.
The seminar brought together 29 researchers in the areas of graph theory, combinatorics, computational geometry, and graph drawing. The common interest was in the exploration of structural properties and the development of algorithms for so-called beyond-planar graphs, i.e., non-planar graphs with topological constraints such as specific types of crossings, or with some forbidden crossing patterns. The seminar began with three introductory talks by experts in the different fields. Abstracts of these talks are collected in this report. Next we discussed and grouped together open research problems about beyond planar graphs, such as their combinatorial structures (e.g, thickness, crossing number, coloring), their topology (e.g., string graph representation), their geometric representations (e.g., straight-line drawing, visibility representation, contact representation), and applications (e.g., algorithms for real-world network visualization). Four working groups were formed and a report from each group is included here.

Sok-Hee Hong, Michael Kaufmann, Stephen G. Kobourov, and János Pach. Beyond-Planar Graphs: Algorithmics and Combinatorics (Dagstuhl Seminar 16452). In Dagstuhl Reports, Volume 6, Issue 11, pp. 35-62, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@Article{hong_et_al:DagRep.6.11.35, author = {Hong, Sok-Hee and Kaufmann, Michael and Kobourov, Stephen G. and Pach, J\'{a}nos}, title = {{Beyond-Planar Graphs: Algorithmics and Combinatorics (Dagstuhl Seminar 16452)}}, pages = {35--62}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2017}, volume = {6}, number = {11}, editor = {Hong, Sok-Hee and Kaufmann, Michael and Kobourov, Stephen G. and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.6.11.35}, URN = {urn:nbn:de:0030-drops-70385}, doi = {10.4230/DagRep.6.11.35}, annote = {Keywords: graph drawing, graph algorithms, graph theory, geometric algorithms, combinatorial geometry, visualization} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

It is proved that the total length of any set of countably many rectifiable curves, whose union meets all straight lines that intersect the unit square U, is at least 2.00002. This is the first improvement on the lower bound of 2 by Jones in 1964. A similar bound is proved for all convex sets U other than a triangle.

Akitoshi Kawamura, Sonoko Moriyama, Yota Otachi, and János Pach. A Lower Bound on Opaque Sets. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 46:1-46:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kawamura_et_al:LIPIcs.SoCG.2016.46, author = {Kawamura, Akitoshi and Moriyama, Sonoko and Otachi, Yota and Pach, J\'{a}nos}, title = {{A Lower Bound on Opaque Sets}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {46:1--46:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.46}, URN = {urn:nbn:de:0030-drops-59386}, doi = {10.4230/LIPIcs.SoCG.2016.46}, annote = {Keywords: barriers; Cauchy-Crofton formula; lower bound; opaque sets} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Following groundbreaking work by Haussler and Welzl (1987), the use of small epsilon-nets has become a standard technique for solving algorithmic and extremal problems in geometry and learning theory. Two significant recent developments are: (i) an upper bound on the size of the smallest epsilon-nets for set systems, as a function of their so-called shallow-cell complexity (Chan, Grant, Konemann, and Sharpe); and (ii) the construction of a set system whose members can be obtained by intersecting a point set in R^4 by a family of half-spaces such that the size of any epsilon-net for them is at least (1/(9*epsilon)) log (1/epsilon) (Pach and Tardos).
The present paper completes both of these avenues of research. We (i) give a lower bound, matching the result of Chan et al., and (ii) generalize the construction of Pach and Tardos to half-spaces in R^d, for any d >= 4, to show that the general upper bound of Haussler and Welzl for the size of the smallest epsilon-nets is tight.

Andrey Kupavskii, Nabil Mustafa, and János Pach. New Lower Bounds for epsilon-Nets. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 54:1-54:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kupavskii_et_al:LIPIcs.SoCG.2016.54, author = {Kupavskii, Andrey and Mustafa, Nabil and Pach, J\'{a}nos}, title = {{New Lower Bounds for epsilon-Nets}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {54:1--54:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.54}, URN = {urn:nbn:de:0030-drops-59467}, doi = {10.4230/LIPIcs.SoCG.2016.54}, annote = {Keywords: epsilon-nets; lower bounds; geometric set systems; shallow-cell complexity; half-spaces} }

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Complete Volume

**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

LIPIcs, Volume 34, SoCG'15, Complete Volume

31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@Proceedings{arge_et_al:LIPIcs.SOCG.2015, title = {{LIPIcs, Volume 34, SoCG'15, Complete Volume}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015}, URN = {urn:nbn:de:0030-drops-52479}, doi = {10.4230/LIPIcs.SOCG.2015}, annote = {Keywords: Analysis of Algorithms and Problem Complexity, Nonnumerical Algorithms and Problems – Geometrical problems and computations, Discrete Mathematics} }

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Front Matter

**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Front Matter, Table of Contents, Preface, Conference Organization

31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. i-xx, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{arge_et_al:LIPIcs.SOCG.2015.i, author = {Arge, Lars and Pach, J\'{a}nos}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {i--xx}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.i}, URN = {urn:nbn:de:0030-drops-50844}, doi = {10.4230/LIPIcs.SOCG.2015.i}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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