# Discrete Methods used in Graph Theory and Linear Programming

Abstract: The content of the thesis is divided into two parts; graph theory and linear programming. The main results in the first part concerns extremal graph theory. Here we want to determine the number of edges in a graph needed to ensure the existence of certain local structures. Upper and lower bounds are given for the number of edges in a graph to ensure triangles and quadrilaterals respectively, when a transition system defined through local edge colourings is taken in account. Partial results on the Erdös-Sós conjecture and Loebl-Komlós-Sós conjecture are given in the affirmative, both for new classes of graphs and for new classes of trees. A proof that every complete graph without monochromatic triangles contains a properly coloured hamiltonian path is also given. In linear programming we study the perceptron algorithm and different modifications in detail, including some geometrical properties of the unit sphere in higher dimensions. An optimal lower bound on the probability that two n-dimensional unit vectors have an inner product of at least 1/sqrt{n} is proved. Modifications to increase the expected time taken by the algorithm with a factor n², compared to earlier algorithms, are made. Also, generalisations of different parts of the modified perceptron algorithm, to make it approximating a maximal margin, are constructed in the thesis.

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