Problemi Turanovog tipa za hipergrafove

master's thesis

• Title (english): Turan type problems for hypergraphs
• Author: Tadej Petar Tukara
• Mentor: Rudi Mrazović (mentor)
• Mentor: Nina Kamčev (sumentor)
• Committee member: Rudi Mrazović (predsjednik povjerenstva)
• Committee member: Nina Kamčev (član povjerenstva)
• Committee member: Vanja Wagner (član povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-09-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu bavili smo se problemima Turanovog tipa za hipergrafove. Prvo smo dokazali klasične rezultate ovog tipa za grafove, Turanov i Erdos-Stone teorem. Zatim smo uveli osnovne pojmove vezano uz hipergrafove. Definirali smo Turanovu gustoću uniformnog hipergrafa i pokazali osnovne rezultate o Turanovoj gustoći degeneriranih hipergrafova, prezasićenosti i napuhavanju. Uveli smo koncept iteriranog napuhavanja i demonstrirali neke konstrukcije koje daju važne donje ograde za Turanovu gustoću. Nakon toga demonstrirali smo neke od metoda za određivanje Turanove gustoće hipergrafova. Uveli smo pojam linka i pomoću njega odredili Turanovu gustoću Fanove ravnine. Također smo pokazali da Turanova gustoća familije hipergrafova može biti manja od Turanove gustoće svakog člana familije i dokazali jednostavnu gornju ogradu na gustoću tijesnog ciklusa duljine 5. Ukratko smo prezentirali glavne ideje dvaju novih rezultata u području koji koriste metodu bojenja sjene hipergrafa i metodu stabilnosti. Zatim smo demonstrirali metodu stabilnosti na primjeru ciklusa i pokazali neke od njenih primjena. Za kraj iznijeli smo nekoliko otvorenih problema Turánovog tipa koji su vezani uz rezultate prezentirane u ovom radu.

Abstract (english)

In this thesis we considered problems of Turaan type for hypergraphs. First we showed the classic results of this type for graphs, Turan’s and Erdos-Stone theorem. Afterwards we introduced the basic notions related to hypergraphs. We defined the Turan density of a uniform hypergraph and showed the basic results on Turan density of degenerate hypergraphs, supersaturation and blowing-up. We introduced the concept of iterated blow-ups and demonstrated some constructions giving important lower bounds on Turan density. Next, we demonstrated some of the methods for determining the Turan density of hypergraphs. We introduced links, and used the link method to determine the Turan density of the Fano plane. We also showed that the Turan density of a family of hypergraphs can be less than the Turan density of each member of the family and proved a simple upper bound on the density of the tight cycle of length 5. We briefly presented the proofs of two new results in this area which use the method of coloring the shadow of a hypergraph and the stability method. We then demonstrated the stability method on the example of a cycle and showed some of its applications. In the end, we presented several open problems of Turan type that are connected to the results presented in this thesis.

• Keywords: Turanov teorem; Erdos-Stone teorem; Turanova gustoća Fanove ravnine
• Keywords (english): Turan’s theorem; Erdos-Stone theorem; Turan density of the Fano plane
• Language: croatian
• URN:NBN: urn:nbn:hr:217:007290
• Study programme: Title: Theoretical Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-02-06 13:47:07