## Extremal problems in the cube and the grid and other combinatorial results

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## Abstract

This dissertation contains results from various areas of combinatorics.

In Chapters 2, 3 and 4 we consider questions in the area of isoperimetric inequalities. In Chapter 2, we find the exact classification of all subsets A⊆{0,1}^n for which both A and A^c minimise the size of the neighbourhood, which answers a question of Aubrun and Szarek. Harper's inequality implies that the initial segments of the simplicial order satisfy these conditions, but we prove that in general there are non-trivial examples of such sets as well.

In Chapter 3, we consider the zero-deletion shadow, which is closely related to the general coordinate deletion shadow introduced by Danh and Daykin. We prove that there is a certain order on [k]^n={0,...,k-1}^n, the n-dimensional grid of side-length k, whose initial segments minimise the size of the zero-deletion shadow.

In Chapter 4, we consider the following generalisation of the Kruskal-Katona theorem on [k]^n. For a set A⊆[k]^n, define the d-shadow of A to be the set of all points x obtained from any y∈A by replacing one non-zero coordinate of y by 0. We find an order on [k]^n whose initial segments minimise the size of the d-shadow.

In Chapter 5, we consider a certain combinatorial game called Toucher-Isolator game that is played on the edges of a given graph G. The value of the game on G measures how many vertices of G one of the players can achieve by using the edges claimed by her. We find the exact value of the game when G is a path or a cycle of a given length, and we prove that among the trees on n vertices, the path on n vertices has the least value of the game. These results improve previous bounds obtained by Dowden, Kang, Mikalački and Stojaković.

In Chapter 6, we consider a problem in Ramsey Theory related to the Hales-Jewett theorem. We prove that for any 2-colouring of [3]^n there exists a monochromatic combinatorial line whose active coordinate set is an interval, provided that n is large. This disproves a conjecture of Conlon and Kamćev.

In Chapter 7, we give a construction of a graph G that is P6-induced-saturated, where P6 is the path on 6 vertices. This answers a question of Axenovich and Csikós.