3 research outputs found
Discrete Macaulay-Steiner Geometry
This thesis is concerned with discrete isoperimetric inequalities and Hilbert functions. Two generalizations of the Ahlswede-Cai local global principle are presented. These results give positive answers to two questions posed by Harper. One of these results is achieved by proving uniqueness of the lexicographic and colexicographic orders in two dimensions. The other result generalizes the technique which is commonly known as compression and includes almost all previously published results in this direction. The Ahlswede-Cai local global principle is a direct corollary of this result. Optimal downsets are studied in rectangles and triangles. All optimal downsets are found. The main result in this direction gives a unified description, optimal downsets are those that are a symmetrization/stabilization of initial segments of the lexicographic and colexicographic orders. Lindsay’s Theorem and the Ahlswede-Katona Edge Isoperimetric Theorem are corollaries. The theory of Macaulay posets is connected to that of Hilbert functions. Several old and new results in both commutative algebra and extremal combinatorics are obtained. Hoefel’s questions on applying Macaulay poset theory to commutative algebra is answered in the affirmative as a by product. A question of Bezrukov and Leck on taking the product of a Macaulay poset with a chain is answered by using a result of Mermin and Peeva. Several answers are given to a problem of Mermin and Peeva
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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 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