## Electronic Theses and Dissertations

751

2012

11-29-2012

Dissertation

#### Degree Name

Doctor of Philosophy

#### Major

Mathematical Sciences

Mathematics

Bela Bollobas

Paul Balister

Anna Kaminska

#### Abstract

This dissertation concerns four problems in combinatorics. In Chapter 2 we consider the Prolonger-Shortener game of F saturation, introduced by Füredi, Reimer and Seress: Players take turns drawing edges on an initially edgeless vertex set of size n with the restriction that they do not complete a copy of a graph in F. The game ends when no more edges can be drawn. Prolonger wants as many edges as possible at the end of the game and Shortener as few as possible. We ask what is the final number of edges with both players playing optimally when F is a fixed path, or a collection of trees. We also consider a directed version of the game. Chapter 3 concerns completion of partial packings of copies of a bipartite graph H. An H-design on n vertices is an edge-disjoint collection of copies of H whose edge sets partition the edge set of the complete graph on n vertices. Given a bipartite graph H and an integer n, let f(n;H) be the smallest integer such that any set of edge-disjoint copies of H on n vertices can be extended to an H-design on at most n+f(n;H) vertices. We establish tight bounds for the growth of f(n;H) as n approaches infinity. In particular, we prove the conjecture of Füredi and Lehel that f(n;H)=o(n). Chapter 4 is dedicated to a particular integer sequence, the Slowgrow sequence, originally introduced by Steven Kalikow. It starts with 1, and having defined terms s1,...,sn the term sn+1 is the smallest positive integer m such that the block sn-m+2...sn+1 has not occurred in the sequence earlier. Our main result is that blocks which can potentially occur multiple times in the sequence actually occur infinitely often. We also prove bounds on the time of the first occurrence of n in the Slowgrow sequence and that the limiting density of every number in the sequence is 0. Chapter 5 is motivated by a question of András Sárközy. We prove sufficient conditions for existence of infinite sets of natural numbers A and B such that the number of solutions of the equation a+b=n where a is in A and b is in B is monotone increasing for n>n0. We also examine a generalized notion of Sidon sets, that is, sets A, B with the property that, for every n>=0, the equation above has at most one solution, i.e., all pairwise sums are distinct.