## Games on Graphs and Other Combinatorial Problems

## Repository URI

## Repository DOI

## Change log

## Authors

## Abstract

In this dissertation, we consider various combinatorial problems. The four chapters after the Introduction concern games on graphs, while latter on, we make progress on some questions in the settings of Rademacher sums and graph theory.

In Chapter 2, we study the

In Chapters 3 and 4, we look at the Waiter-Client

In Chapter 3, we determine the duration of the game under the optimal play of both players when

In Chapter 5, we consider the so-called restricted online Ramsey numbers, which correspond to a certain colouring game in the Builder-Painter setup. We provide a tight lower bound for the restricted online Ramsey numbers of matchings as long as the number of the allowed colours is small, resolving the conjecture of Briggs and Cox.

The setting in the next two chapters is the following. Set

In Chapter 6, we make progress towards an old conjecture of Tomaszewski, which concerns concentration of such random variables

In Chapter 8, we obtain the best possible bounds for the following problem, first studied by Erd\H{o}s, Pach, Pollak and Tuza: given a connected, triangle-free graph on

In Chapter 9, we construct

Finally, in Chapter 10, we obtain a result about the existence of the antipodal paths with few colour changes in a two-colouring of the edges of the hypercube graph.