Document Open Access Logo

Ramsey Properties of Randomly Perturbed Hypergraphs

Authors Elad Aigner-Horev , Dan Hefetz , Mathias Schacht

PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2024.59.pdf
  • Filesize: 0.81 MB
  • 18 pages

Document Identifiers

Author Details

Elad Aigner-Horev
  • School of Computer Science, Ariel University, Israel
Dan Hefetz
  • School of Computer Science, Ariel University, Israel
Mathias Schacht
  • Fachbereich Mathematik, Universität Hamburg, Germany

Cite AsGet BibTex

Elad Aigner-Horev, Dan Hefetz, and Mathias Schacht. Ramsey Properties of Randomly Perturbed Hypergraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 59:1-59:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.59

Abstract

We study Ramsey properties of randomly perturbed 3-uniform hypergraphs. For t ≥ 2, write K^(3)_t to denote the 3-uniform expanded clique hypergraph obtained from the complete graph K_t by expanding each of the edges of the latter with a new additional vertex. For an even integer t ≥ 4, let M denote the asymmetric maximal density of the pair (K^(3)_t, K^(3)_{t/2}). We prove that adding a set F of random hyperedges satisfying |F| ≫ n^{3-1/M} to a given n-vertex 3-uniform hypergraph H with non-vanishing edge density asymptotically almost surely results in a perturbed hypergraph enjoying the Ramsey property for K^(3)_t and two colours. We conjecture that this result is asymptotically best possible with respect to the size of F whenever t ≥ 6 is even. The key tools of our proof are a new variant of the hypergraph regularity lemma accompanied with a tuple lemma providing appropriate control over joint link graphs. Our variant combines the so called strong and the weak hypergraph regularity lemmata.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Random graphs
  • Mathematics of computing → Hypergraphs
Keywords
  • Ramsey Theory
  • Smoothed Analysis
  • Random Hypergraphs

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. E. Aigner-Horev, O. Danon, D. Hefetz, and S. Letzter. Large rainbow cliques in randomly perturbed dense graphs. SIAM J. Discrete Math., 36(4):2975-2994, 2022. Google Scholar
  2. E. Aigner-Horev, O. Danon, D. Hefetz, and S. Letzter. Small rainbow cliques in randomly perturbed dense graphs. European J. Combin., 101:Paper No. 103452, 34 pages, 2022. Google Scholar
  3. E. Aigner-Horev and D. Hefetz. Rainbow hamilton cycles in randomly colored randomly perturbed dense graphs. SIAM J. Discrete Math., 35(3):1569-1577, 2021. Google Scholar
  4. E. Aigner-Horev, D. Hefetz, and M. Krivelevich. Minors, connectivity, and diameter in randomly perturbed sparse graphs. Arxiv preprint, 2022. URL: https://arxiv.org/abs/2212.07192.
  5. E. Aigner-Horev, D. Hefetz, and M. Krivelevich. Cycle lengths in randomly perturbed graphs. Random Structures & Algorithms, 63(4):867-884, 2023. Google Scholar
  6. E. Aigner-Horev, D. Hefetz, and A. Lahiri. Rainbow trees in uniformly edge-colored graphs. Random Structures & Algorithms, 62(2):287-303, 2023. Google Scholar
  7. E. Aigner-Horev, D. Hefetz, and M. Schacht. Ramsey properties of randomly perturbed hypergraphs. Arxiv preprint, 2024. URL: https://arxiv.org/abs/2311.01750.
  8. E. Aigner-Horev and Y. Person. Monochromatic Schur triples in randomly perturbed dense sets of integers. SIAM J. Discrete Math., 33(4):2175-2180, 2019. Google Scholar
  9. P. Allen, O. Parczyk, and V. Pfenninger. Resilience for tight Hamiltonicity. Arxiv preprint, 2021. URL: https://arxiv.org/abs/2105.04513.
  10. J. Balogh, A. Treglown, and A. Z. Wagner. Tilings in randomly perturbed dense graphs. Combin. Probab. Comput., 28(2):159-176, 2019. Google Scholar
  11. W. Bedenknecht, J. Han, Y. Kohayakawa, and G. O. Mota. Powers of tight hamilton cycles in randomly perturbed hypergraphs. Random Structures & Algorithms, 55(4):795-807, 2019. Google Scholar
  12. J. Böttcher, J. Han, Y. Kohayakawa, R. Montgomery, O. Parczyk, and Y. Person. Universality for bounded degree spanning trees in randomly perturbed graphs. Random Structures & Algorithms, 55(4):854-864, 2019. Google Scholar
  13. J. Böttcher, R. Montgomery, O. Parczyk, and Y. Person. Embedding spanning bounded degree graphs in randomly perturbed graphs. Mathematika, 66(2):422-447, 2020. Google Scholar
  14. C. Bowtell, R. Hancock, and J. Hyde. Proof of the Kohayakawa-Kreuter conjecture for the majority of cases. Arxiv preprint, 2023. URL: https://arxiv.org/abs/2307.16760.
  15. M. Christoph, A. Martinsson, R. Steiner, and Y. Wigderson. Resolution of the Kohayakawa-Kreuter conjecture. Arxiv preprint, 2024. URL: https://arxiv.org/abs/2402.03045.
  16. D. Conlon, J. Fox, and V. Rödl. Hedgehogs are not colour blind. J. Combinatorics, 8(3):475-485, 2017. Google Scholar
  17. D. Conlon and W. T. Gowers. Combinatorial theorems in sparse random sets. Ann. of Math., 184(2):367-454, 2016. Google Scholar
  18. S. Das, C. Knierim, and P. Morris. Schur properties of randomly perturbed sets. Arxiv preprint, 2022. URL: https://arxiv.org/abs/2205.01456.
  19. S. Das, P. Morris, and A. Treglown. Vertex Ramsey properties of randomly perturbed graphs. Random Structures & Algorithms, 57(4):983-1006, 2020. Google Scholar
  20. S. Das and A. Treglown. Ramsey properties of randomly perturbed graphs: cliques and cycles. Combin. Probab. Comput., 29(6):830-867, 2020. Google Scholar
  21. A. Dudek, C. Reiher, A. Ruciński, and M. Schacht. Powers of Hamiltonian cycles in randomly augmented graphs. Random Structures & Algorithms, 56(1):122-141, 2020. Google Scholar
  22. P. Erdős. Some theorems on graphs. Riveon Lematematika, 9:13-17, 1955. Google Scholar
  23. P. Erdős. On a theorem of rademacher-turán. Illinois J. Math., 6:122-127, 1962. Google Scholar
  24. P. Erdős. On the number of complete subgraphs contained in certain graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl., 7:459-464, 1962. Google Scholar
  25. P. Frankl and V. Rödl. Extremal problems on set systems. Random Structures & Algorithms, 20(2):131-164, 2002. Google Scholar
  26. E. Friedgut, V. Rödl, and M. Schacht. Ramsey properties of random discrete structures. Random Structures & Algorithms, 37(4):407-436, 2010. Google Scholar
  27. L. Gugelmann, R. Nenadov, Y. Person, N. Škorić, A. Steger, and H. Thomas. Symmetric and asymmetric ramsey properties in random hypergraphs. Forum Math. Sigma, 5:Paper No. e28, 47 pages, 2017. Google Scholar
  28. J. Han and Y. Zhao. Hamiltonicity in randomly perturbed hypergraphs. J. Combin. Theory Ser. B, 144:14-31, 2020. Google Scholar
  29. J. Hyde. Towards the 0-statement of the Kohayakawa-Kreuter conjecture. Arxiv preprint, 2021. URL: https://arxiv.org/abs/2105.15151.
  30. Y. Kohayakawa and B. Kreuter. Threshold functions for asymmetric ramsey properties involving cycles. Random Structures & Algorithms, 11(3):245-276, 1997. Google Scholar
  31. Y. Kohayakawa, M. Schacht, and R. Spöhel. Upper bounds on probability thresholds for asymmetric Ramsey properties. Random Structures & Algorithms, 44(1):1-28, 2014. Google Scholar
  32. J. Komlós, M. Shokoufandeh, A. Simonovits, and E. Szemerédi. The regularity lemma and its applications in graph theory. In Theoretical aspects of computer science, volume 2292 of Lecture Notes in Comput. Sci., pages 84-112. Springer, Berlin, 2002. Google Scholar
  33. J. Komlós and M. Simonovits. Szemerédi’s regularity lemma and its applications in graph theory. In Combinatorics, Paul Erdős is eighty, Vol. 2, volume 2 of Bolyai Soc. Math. Stud., 1996. Google Scholar
  34. M. Krivelevich, M. Kwan, and B. Sudakov. Cycles and matchings in randomly perturbed digraphs and hypergraphs. Combin. Probab. Comput., 25(6):909-927, 2016. Google Scholar
  35. M. Krivelevich, M. Kwan, and B. Sudakov. Bounded-degree spanning trees in randomly perturbed graphs. SIAM J. Discrete Math., 31(1), 2017. Google Scholar
  36. M. Krivelevich, B. Sudakov, and Prasad Tetali. On smoothed analysis in dense graphs and formulas. Random Structures & Algorithms, 29(2):180-193, 2006. Google Scholar
  37. E. Kuperwasser, W. Samotij, and Y. Wigderson. On the kohayakawa-kreuter conjecture. Arxiv preprint, 2023. URL: https://arxiv.org/abs/2307.16611.
  38. A. Liebenau, L. Mattos, W. Mendonça, and J. Skokan. Asymmetric Ramsey properties of random graphs involving cliques and cycles. Random Structures & Algorithms, 62(4):1035-1055, 2023. Google Scholar
  39. T. Łuczak, A. Ruciński, and B. Voigt. Ramsey properties of random graphs. J. Combin. Theory Ser. B, 56(1):55-68, 1992. Google Scholar
  40. M. Marciniszyn, J. Skokan, , R. Spöhel, and A. Steger. Asymmetric Ramsey properties of random graphs involving cliques. Random Structures & Algorithms, 34(4):419-453, 2009. Google Scholar
  41. A. McDowell and R. Mycroft. Hamilton 𝓁-cycles in randomly perturbed hypergraphs. Electron. J. Combin., 25(4):Paper No. 4.36, 30 pages, 2018. Google Scholar
  42. F. Mousset, R. Nenadov, and W. Samotij. Towards the Kohayakawa-Kreuter conjecture on asymmetric Ramsey properties. Combin. Probab. Comput., 29(6):943-955, 2020. Google Scholar
  43. D. Mubayi. A hypergraph extension of turán’s theorem. J. Combin. Theory Ser. B, 96(1):122-134, 2006. Google Scholar
  44. R. Nenadov, Y. Person, N. Škorić, and A. Steger. An algorithmic framework for obtaining lower bounds for random Ramsey problems. J. Combin. Theory Ser. B, 124:1-38, 2017. Google Scholar
  45. Rajko. Nenadov and A. Steger. A short proof of the random Ramsey theorem. Combin. Probab. Comput., 25(1):130-144, 2016. Google Scholar
  46. E. Powierski. Ramsey properties of randomly perturbed dense graphs. Arxiv preprint, 2019. URL: https://arxiv.org/abs/1902.02197.
  47. V. Rödl and A. Ruciński. Lower bounds on probability thresholds for ramsey properties. In Combinatorics, Paul Erdős is eighty, Vol. 1, Bolyai Soc. Math. Stud., pages 317-346. János Bolyai Math. Soc., Budapest, 1993. Google Scholar
  48. V. Rödl and A. Ruciński. Random graphs with monochromatic triangles in every edge coloring. Random Structures & Algorithms, 5(2):253-270, 1994. Google Scholar
  49. V. Rödl and A. Ruciński. Threshold functions for Ramsey properties. J. Amer. Math. Soc., 8(4):917-942, 1995. Google Scholar
  50. V. Rödl and A. Ruciński. Ramsey properties of random hypergraphs. J. Combin. Theory Ser. A, 81(1):1-33, 1998. Google Scholar
  51. V. Rödl and M. Schacht. Regular partitions of hypergraphs: regularity lemmas. Combin. Probab. Comput., 16(6):833-885, 2007. Google Scholar
  52. D. Spielman and S. Teng. Smoothed Analysis of algorithms: why the simplex algorithm usually takes polynomial time. Journal of the ACM, 51:385-463, 2004. Google Scholar
  53. D. Spielman and S. Teng. Smoothed analysis: an attempt to explain the behavior of algorithms in practice. Communications of the ACM, 52:76-84, 2009. Google Scholar
  54. E. Szemerédi. Regular partitions of graphs. In Problèmes combinatoires et théorie des graphes, volume 260 of Colloq. Internat. CNRS, pages 399-401. CNRS, Paris, 1978. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact