Term of Award
Spring 2024
Degree Name
Master of Science in Mathematics (M.S.)
Document Type and Release Option
Thesis (open access)
Copyright Statement / License for Reuse
This work is licensed under a Creative Commons Attribution 4.0 License.
Department
Department of Mathematical Sciences
Committee Chair
Hua Wang
Committee Member 1
Sungkon Chang
Committee Member 2
Daniel Gray
Abstract
In the first part of this work we introduce a variation of the well-known Ramsey number, which we call the Exact Ramsey Number (ERN) or Strong Ramsey Number (STN). The ERN of $s$ and $t$, denoted by $\mathcal{R}(s,t)$, is defined as the minimum $\ell$ such that any two colouring of $K_{\ell}$ that does not contain monochromatic $K_{s+1}$ of the first colour or monochromatic $K_{t+1}$ of the second colour must contain a monochromatic $K_s$ of the first colour and a monochromatic $K_t$ of the second colour. We establish some properties of these ERNs, including the evaluation of small ERNs and generalizations of well-known theories of the classic Ramsey numbers. We also study the relation between the ERN and the classic Ramsey numbers.
The second part of this work is on the Middle Levels Problem. The middle level conjecture is about the presence of a Hamiltonian cycle in the middle layer graph of a hypercube. In this work we give a new approach to the Middle Levels Problem. This approach can also be extended to the Central Levels Problem.
Recommended Citation
Mutze, Proof of the middle levels conjecture, Proc. Lond. Math. Soc. 112(4) (2016), 677–713. Gregor, O. Micka, and T. Mutze, On the central levels problem, 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, LIPIcs, vol. 168, 2020, pp. 60:1–60:17 P. Gregor and R. ˇSkrekovski. On generalized middle-level problem. Inform. Sci., 180(12):2448–2457, 2010
Research Data and Supplementary Material
No
