Extremal results in hypergraph theory via the absorption method

Sanhueza Matamala, Nicolás (2020). Extremal results in hypergraph theory via the absorption method. University of Birmingham. Ph.D.

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Abstract

The so-called "absorbing method" was first introduced in a systematic way by Rödl, Ruciński and Szemerédi in 2006, and has found many uses ever since. Speaking in a general sense, it is useful for finding spanning substructures of combinatorial structures. We establish various results of different natures, in both graph and hypergraph theory, most of them using the absorbing method:

1. We prove an asymptotically best-possible bound on the strong chromatic number with respect to the maximum degree of the graph. This establishes a weak version of a conjecture of Aharoni, Berger and Ziv.
2. We determine asymptotic minimum codegree thresholds which ensure the existence of tilings with tight cycles (of a given size) in uniform hypergraphs. Moreover, we prove results on coverings with tight cycles.
3. We show that every 2-coloured complete graph on the integers contains a monochromatic infinite path whose vertex set is sufficiently "dense" in the natural numbers. This improves results of Galvin and Erdős and of DeBiasio and McKenney.

Type of Work: Thesis (Doctorates > Ph.D.)
Award Type: Doctorates > Ph.D.
Supervisor(s):
Supervisor(s)EmailORCID
Lo, Allans.a.lo@bham.ac.ukUNSPECIFIED
Kühn, Danielad.kuhn@bham.ac.ukUNSPECIFIED
Licence: All rights reserved
College/Faculty: Colleges (2008 onwards) > College of Engineering & Physical Sciences
School or Department: School of Mathematics
Funders: Other
Other Funders: Becas Chile scholarship scheme from CONICYT, Chile
Subjects: Q Science > QA Mathematics
URI: http://etheses.bham.ac.uk/id/eprint/9999

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