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## Turán and Ramsey numbers for 3-uniform hyperpaths

### Eliza Jackowska-Boryc

#### Abstract

The main subject of this dissertation are Ramsey and Turán numbers for the 3-uniform loose path P of length 3. We began by introducing the basic terminology and our results related to Ramsey and Turán numbers. In chapter 1, we formulated the most crucial result - Theorem 2.2. It gives the exact value for Ramsey number R(P;r)=r+6, for a number of colors less than 8. One of the most important method in determining Ramsey numbers is an application of Turán numbers. In order to prove Theorem 2.2. we had to use Erdős-Ko-Rado Theorem and Theorem related to Turán number for a triangle. The most important Turán-type Theorem is 2.7, which determines Turán number for P, for all n. Moreover, in chapter 2, we stated Theorems 2.9 and 2.10, which were extensions of a standard approach of Turán number- Turán numbers of the second and the third order. Those Theorems appeared to be very useful in the proof of Theorem 2.2, which was introduced in chapter 3. In chapter 4, w focused on conditional Turán numbers and their applications in the proofs of Theorem 2.7. The last chapter 5 was devoted to the presentation of the proofs of remaining theorems.
Record ID
UAM6db86f6837e34277b79e022763352e00
Diploma type
Doctor of Philosophy
Author
Eliza Jackowska-Boryc (SNŚ/WMiI/FoMaCS) Eliza Jackowska-Boryc,,
Title in Polish
Liczby Turána i Ramseya dla 3-jednolitych ścieżek
Title in English
Turán and Ramsey numbers for 3-uniform hyperpaths
Language
en English
Certifying Unit
Faculty of Mathematics and Computer Science (SNŚ/WMiI/FoMaCS)
Discipline
mathematics / (mathematics domain) / (physical sciences)
Scientific discipline (2.0)
6.3 mathematics
Defense Date
16-11-2018
Title date
16-11-2018
Supervisor
URL
http://hdl.handle.net/10593/24111 opening in a new tab
Keywords in English
Turán number, Ramsey number, hypergraph, path
Abstract in English
The main subject of this dissertation are Ramsey and Turán numbers for the 3-uniform loose path P of length 3. We began by introducing the basic terminology and our results related to Ramsey and Turán numbers. In chapter 1, we formulated the most crucial result - Theorem 2.2. It gives the exact value for Ramsey number R(P;r)=r+6, for a number of colors less than 8. One of the most important method in determining Ramsey numbers is an application of Turán numbers. In order to prove Theorem 2.2. we had to use Erdős-Ko-Rado Theorem and Theorem related to Turán number for a triangle. The most important Turán-type Theorem is 2.7, which determines Turán number for P, for all n. Moreover, in chapter 2, we stated Theorems 2.9 and 2.10, which were extensions of a standard approach of Turán number- Turán numbers of the second and the third order. Those Theorems appeared to be very useful in the proof of Theorem 2.2, which was introduced in chapter 3. In chapter 4, w focused on conditional Turán numbers and their applications in the proofs of Theorem 2.7. The last chapter 5 was devoted to the presentation of the proofs of remaining theorems.
Thesis file

Uniform Resource Identifier
https://researchportal.amu.edu.pl/info/phd/UAM6db86f6837e34277b79e022763352e00/

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