Ramseyeva teorija : diplomsko delo

Hočevar, Mitja (author), Šparl, Primož (mentor)

Tema diplomskega dela sodi na področje teorije grafov. Ramseyeva teorija se v njeni najosnovnejši obliki ukvarja z določanjem minimalnega naravnega števila, da polni graf tega reda ob poljubnem barvanju njegovih povezav z danim številom barv vsebuje podgraf vnaprej določene velikosti, ki ima vse povezave iste barve. Gre za tako imenovana Ramseyeva števila. V diplomskem delu predstavimo osnovne pojme teorije grafov, ki so potrebni za razumevanje izrekov Ramseyeve teorije in njihovih dokazov. Predstavimo osnovni izrek o obstoju Ramseyevih števil za poljubne pozitivne rede monokromatičnih klik. Obravnavamo problem določitve zgornje meje za Ramseyeva števila. Predstavimo do danes določena Ramseyeva števila in za nekatera tudi dokažemo, da so res ustrezna. Prikažemo tudi nekaj možnosti uporabe Ramseyeve teorije pri konkretnih problemih.

• Language: Slovenian
• Keywords: teorija grafov, polni graf, Ramseyevo število
• Work type: Bachelor thesis/paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: PEF - Faculty of Education
• Publisher: [M. Hočevar]
• Year: 2015
• Number of pages: 29 str.
• PID: 20.500.12556/RUL-72659
• UDC: 519.17(043.2)
• COBISS.SI-ID: 10702409
• Publication date in RUL: 29.09.2015

HOČEVAR, Mitja, 2015, Ramseyeva teorija : diplomsko delo [online]. Bachelor’s thesis. M. Hočevar. [Accessed 14 March 2025]. Retrieved from: http://pefprints.pef.uni-lj.si/id/eprint/3053

Secondary language

• Language: English
• Title: Ramsey theory

Abstract

This BCs thesis deals with topics from graph theory. Ramsey theory in its most basic form deals with the problem of determining the minimal positive integer, such that for any edge-coloring of the complete graph of this size with a prescribed number of colors one can find a subgraph of predefined size all of whose edges are of the same colour. These minimal sizes are called Ramsey numbers. In this BCs thesis we present basic notions of graph theory needed to understand the basic theorem of Ramsey theory and its proof. We discuss the problem of determining bounds for Ramsey numbers. We present Ramsey numbers found till today, and for some of them we prove that they are correct. We indicate some possibilities of how to use Ramsey theory with a few examples.

• Keywords: graph theory