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Documents authored by Pálvölgyi, Dömötör

Document
DOI: 10.4230/LIPIcs.SoCG.2022.32
Three-Chromatic Geometric Hypergraphs

Authors: Gábor Damásdi and Dömötör Pálvölgyi

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

Abstract
We prove that for any planar convex body C there is a positive integer m with the property that any finite point set P in the plane can be three-colored such that there is no translate of C containing at least m points of P, all of the same color. As a part of the proof, we show a strengthening of the Erdős-Sands-Sauer-Woodrow conjecture. Surprisingly, the proof also relies on the two dimensional case of the Illumination conjecture.
Cite as

Gábor Damásdi and Dömötör Pálvölgyi. Three-Chromatic Geometric Hypergraphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{damasdi_et_al:LIPIcs.SoCG.2022.32,
  author =	{Dam\'{a}sdi, G\'{a}bor and P\'{a}lv\"{o}lgyi, D\"{o}m\"{o}t\"{o}r},
  title =	{{Three-Chromatic Geometric Hypergraphs}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{32:1--32:13},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.32},
  URN =		{urn:nbn:de:0030-drops-160401},
  doi =		{10.4230/LIPIcs.SoCG.2022.32},
  annote =	{Keywords: Discrete geometry, Geometric hypergraph coloring, Decomposition of multiple coverings}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.47
Almost-Monochromatic Sets and the Chromatic Number of the Plane

Authors: Nóra Frankl, Tamás Hubai, and Dömötör Pálvölgyi

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

Abstract
In a colouring of ℝ^d a pair (S,s₀) with S ⊆ ℝ^d and with s₀ ∈ S is almost-monochromatic if S⧵{s₀} is monochromatic but S is not. We consider questions about finding almost-monochromatic similar copies of pairs (S,s₀) in colourings of ℝ^d, ℤ^d, and of ℚ under some restrictions on the colouring. Among other results, we characterise those (S,s₀) with S ⊆ ℤ for which every finite colouring of ℝ without an infinite monochromatic arithmetic progression contains an almost-monochromatic similar copy of (S,s₀). We also show that if S ⊆ ℤ^d and s₀ is outside of the convex hull of S⧵{s₀}, then every finite colouring of ℝ^d without a monochromatic similar copy of ℤ^d contains an almost-monochromatic similar copy of (S,s₀). Further, we propose an approach based on finding almost-monochromatic sets that might lead to a human-verifiable proof of χ(ℝ²) ≥ 5.
Cite as

Nóra Frankl, Tamás Hubai, and Dömötör Pálvölgyi. Almost-Monochromatic Sets and the Chromatic Number of the Plane. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 47:1-47:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{frankl_et_al:LIPIcs.SoCG.2020.47,
  author =	{Frankl, N\'{o}ra and Hubai, Tam\'{a}s and P\'{a}lv\"{o}lgyi, D\"{o}m\"{o}t\"{o}r},
  title =	{{Almost-Monochromatic Sets and the Chromatic Number of the Plane}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{47:1--47:15},
  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.2020.47},
  URN =		{urn:nbn:de:0030-drops-122054},
  doi =		{10.4230/LIPIcs.SoCG.2020.47},
  annote =	{Keywords: discrete geometry, Hadwiger-Nelson problem, Euclidean Ramsey theory}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.60
Radon Numbers Grow Linearly

Authors: Dömötör Pálvölgyi

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

Abstract
Define the k-th Radon number r_k of a convexity space as the smallest number (if it exists) for which any set of r_k points can be partitioned into k parts whose convex hulls intersect. Combining the recent abstract fractional Helly theorem of Holmsen and Lee with earlier methods of Bukh, we prove that r_k grows linearly, i.e., r_k ≤ c(r₂)⋅ k.
Cite as

Dömötör Pálvölgyi. Radon Numbers Grow Linearly. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 60:1-60:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{palvolgyi:LIPIcs.SoCG.2020.60,
  author =	{P\'{a}lv\"{o}lgyi, D\"{o}m\"{o}t\"{o}r},
  title =	{{Radon Numbers Grow Linearly}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{60:1--60:5},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.60},
  URN =		{urn:nbn:de:0030-drops-122183},
  doi =		{10.4230/LIPIcs.SoCG.2020.60},
  annote =	{Keywords: discrete geometry, convexity space, Radon number}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2017.47
Proper Coloring of Geometric Hypergraphs

Authors: Balázs Keszegh and Dömötör Pálvölgyi

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

Abstract
We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored such that any member of F that contains at least m points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then m=3 is sufficient. We prove that when F is the family of all homothetic copies of a given convex polygon, then such an m exists. We also study the problem in higher dimensions.
Cite as

Balázs Keszegh and Dömötör Pálvölgyi. Proper Coloring of Geometric Hypergraphs. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 47:1-47:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{keszegh_et_al:LIPIcs.SoCG.2017.47,
  author =	{Keszegh, Bal\'{a}zs and P\'{a}lv\"{o}lgyi, D\"{o}m\"{o}t\"{o}r},
  title =	{{Proper Coloring of Geometric Hypergraphs}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{47:1--47:15},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.47},
  URN =		{urn:nbn:de:0030-drops-71926},
  doi =		{10.4230/LIPIcs.SoCG.2017.47},
  annote =	{Keywords: discrete geometry, decomposition of multiple coverings, geometric hypergraph coloring}
}
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