7 research outputs found
Intervals in the Hales-Jewett theorem
The Hales-Jewett theorem states that for any and there exists an
such that any -colouring of the elements of contains a monochromatic
combinatorial line. We study the structure of the wildcard set which determines this monochromatic line, showing that when is odd
there are -colourings of where the wildcard set of a monochromatic
line cannot be the union of fewer than intervals. This is tight, as for
sufficiently large there are always monochromatic lines whose wildcard set is
the union of at most intervals.Comment: 4 page
Intervals in the Hales-Jewett theorem
The Hales-Jewett theorem states that for any m and r there exists an n such that any r-colouring of the elements of [m]^n contains a monochromatic combinatorial line. We study the structure of the wildcard set S ā [n] which determines this monochromatic line, showing that when r is odd there are r-colourings of [3]^n where the wildcard set of a monochromatic line cannot be the union of fewer than r intervals. This is tight, as for n sufficiently large there are always monochromatic lines whose wildcard set is the union of at most r intervals
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Extremal problems in the cube and the grid and other combinatorial results
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 Aā{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 Aā[k]^n, define the d-shadow of A to be the set of all points x obtained from any yāA 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, MikalaÄki and StojakoviÄ.
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 KamÄev.
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 CsikĆ³s