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

The colorful Helly theorem and Tverberg’s theorem are fundamental results in discrete geometry. We prove a theorem which interpolates between the two. In particular, we show the following for any integers d ≥ m ≥ 1 and k a prime power. Suppose F₁, F₂, … , F_m are families of convex sets in ℝ^d, each of size n > (d/m+1)(k-1), such that for any choice C_i ∈ F_i we have ⋂_{i = 1}^m C_i ≠ ∅. Then, one of the families F_i admits a Tverberg k-partition. That is, one of the F_i can be partitioned into k nonempty parts such that the convex hulls of the parts have nonempty intersection. As a corollary, we also obtain a result concerning r-dimensional transversals to families of convex sets in ℝ^d that satisfy the colorful Helly hypothesis, which extends the work of Karasev and Montejano.

Michael Gene Dobbins, Andreas F. Holmsen, and Dohyeon Lee. Colorful Intersections and Tverberg Partitions. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 52:1-52:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dobbins_et_al:LIPIcs.SoCG.2024.52, author = {Dobbins, Michael Gene and Holmsen, Andreas F. and Lee, Dohyeon}, title = {{Colorful Intersections and Tverberg Partitions}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {52:1--52:13}, 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.2024.52}, URN = {urn:nbn:de:0030-drops-199973}, doi = {10.4230/LIPIcs.SoCG.2024.52}, annote = {Keywords: Tverberg’s theorem, geometric transversals, topological combinatorics, configuration space/test map, discrete Morse theory} }

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

Intersection patterns of convex sets in ℝ^d have the remarkable property that for d+1 ≤ k ≤ 𝓁, in any sufficiently large family of convex sets in ℝ^d, if a constant fraction of the k-element subfamilies have nonempty intersection, then a constant fraction of the 𝓁-element subfamilies must also have nonempty intersection. Here, we prove that a similar phenomenon holds for any topological set system ℱ in ℝ^d. Quantitatively, our bounds depend on how complicated the intersection of 𝓁 elements of ℱ can be, as measured by the maximum of the ⌈d/2⌉ first Betti numbers. As an application, we improve the fractional Helly number of set systems with bounded topological complexity due to the third author, from a Ramsey number down to d+1. We also shed some light on a conjecture of Kalai and Meshulam on intersection patterns of sets with bounded homological VC dimension. A key ingredient in our proof is the use of the stair convexity of Bukh, Matoušek and Nivasch to recast a simplicial complex as a homological minor of a cubical complex.

Xavier Goaoc, Andreas F. Holmsen, and Zuzana Patáková. A Stepping-Up Lemma for Topological Set Systems. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{goaoc_et_al:LIPIcs.SoCG.2021.40, author = {Goaoc, Xavier and Holmsen, Andreas F. and Pat\'{a}kov\'{a}, Zuzana}, title = {{A Stepping-Up Lemma for Topological Set Systems}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {40:1--40:17}, 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.2021.40}, URN = {urn:nbn:de:0030-drops-138396}, doi = {10.4230/LIPIcs.SoCG.2021.40}, annote = {Keywords: Helly-type theorem, Topological combinatorics, Homological minors, Stair convexity, Cubical complexes, Homological VC dimension, Ramsey-type theorem} }

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

We study the problem of deciding if a given triple of permutations can be realized as geometric permutations of disjoint convex sets in R^3. We show that this question, which is equivalent to deciding the emptiness of certain semi-algebraic sets bounded by cubic polynomials, can be "lifted" to a purely combinatorial problem. We propose an effective algorithm for that problem, and use it to gain new insights into the structure of geometric permutations.

Xavier Goaoc, Andreas Holmsen, and Cyril Nicaud. An Experimental Study of Forbidden Patterns in Geometric Permutations by Combinatorial Lifting. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 40:1-40:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{goaoc_et_al:LIPIcs.SoCG.2019.40, author = {Goaoc, Xavier and Holmsen, Andreas and Nicaud, Cyril}, title = {{An Experimental Study of Forbidden Patterns in Geometric Permutations by Combinatorial Lifting}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {40:1--40:16}, 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.2019.40}, URN = {urn:nbn:de:0030-drops-104442}, doi = {10.4230/LIPIcs.SoCG.2019.40}, annote = {Keywords: Geometric permutation, Emptiness testing of semi-algebraic sets, Computer-aided proof} }

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

We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial complexity of the bodies and the topological complexity of their realization space. On one hand, we show that every combinatorial type can be realized by an arrangement of convex bodies and (under mild assumptions) its realization space is contractible. On the other hand, we prove a universality theorem that says that the restriction of the realization space to arrangements of convex polygons with a bounded number of vertices can have the homotopy type of any primary semialgebraic set.

Michael Gene Dobbins, Andreas Holmsen, and Alfredo Hubard. Realization Spaces of Arrangements of Convex Bodies. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 599-614, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{dobbins_et_al:LIPIcs.SOCG.2015.599, author = {Dobbins, Michael Gene and Holmsen, Andreas and Hubard, Alfredo}, title = {{Realization Spaces of Arrangements of Convex Bodies}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {599--614}, 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.2015.599}, URN = {urn:nbn:de:0030-drops-51020}, doi = {10.4230/LIPIcs.SOCG.2015.599}, annote = {Keywords: Oriented matroids, Convex sets, Realization spaces, Mnev’s universality theorem} }

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