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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 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 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} }

Document

**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 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We propose a new distribution-free model of social networks. Our definitions are motivated by one of the most universal signatures of social networks, triadic closure - the property that pairs of vertices with common neighbors tend to be adjacent. Our most basic definition is that of a c-closed graph, where for every pair of vertices u,v with at least c common neighbors, u and v are adjacent. We study the classic problem of enumerating all maximal cliques, an important task in social network analysis. We prove that this problem is fixed-parameter tractable with respect to c on c-closed graphs. Our results carry over to weakly c-closed graphs, which only require a vertex deletion ordering that avoids pairs of non-adjacent vertices with c common neighbors. Numerical experiments show that well-studied social networks tend to be weakly c-closed for modest values of c.

Jacob Fox, Tim Roughgarden, C. Seshadhri, Fan Wei, and Nicole Wein. Finding Cliques in Social Networks: A New Distribution-Free Model. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fox_et_al:LIPIcs.ICALP.2018.55, author = {Fox, Jacob and Roughgarden, Tim and Seshadhri, C. and Wei, Fan and Wein, Nicole}, title = {{Finding Cliques in Social Networks: A New Distribution-Free Model}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {55:1--55:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.55}, URN = {urn:nbn:de:0030-drops-90596}, doi = {10.4230/LIPIcs.ICALP.2018.55}, annote = {Keywords: Graph algorithms, social networks, fixed-parameter tractability} }

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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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Invited Talk

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

Many problems in discrete and computational geometry can be viewed as finding patterns in graphs or hypergraphs which arise from geometry or algebra. Famous Ramsey, Turán, and Szemerédi-type results prove the existence of certain patterns in graphs and hypergraphs under mild assumptions. We survey recent results which show much stronger/larger patterns for graphs and hypergraphs that arise from geometry or algebra. We further discuss whether the stronger results in these settings are due to geometric, algebraic, combinatorial, or topological properties of the graphs.

Jacob Fox. Discrete Geometry, Algebra, and Combinatorics (Invited Talk). In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fox:LIPIcs.SoCG.2016.2, author = {Fox, Jacob}, title = {{Discrete Geometry, Algebra, and Combinatorics}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {2:1--2:1}, 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.2016.2}, URN = {urn:nbn:de:0030-drops-58948}, doi = {10.4230/LIPIcs.SoCG.2016.2}, annote = {Keywords: discrete geometry, extremal combinatorics, regularity lemmas, Ramsey theory} }

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