## A Collection of Problems in Extremal Combinatorics

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Extremal combinatorics is concerned with how large or small a combinatorial
structure can be if we insist it satis es certain properties. In this thesis we
investigate four different problems in extremal combinatorics, each with its
own unique
flavour.
We begin by examining a graph saturation problem. We say a graph G
is H-saturated if G contains no copy of H as a subgraph, but the addition
of any new edge to G creates a copy of H. We look at how few edges a Kp-
saturated graph can have when we place certain conditions on its minimum
degree.
We look at a problem in Ramsey Theory. The k-colour Ramsey number
Rk(H) of a graph H is de ned as the least integer n such that every k-
colouring of Kn contains a monochromatic copy of H. For an integer r > 3
let Cr denote the cycle on r vertices. By studying a problem related to
colourings without short odd cycles, we prove new lower bounds for Rk(Cr)
when r is odd.
Bootstrap percolation is a process in graphs that can be used to model
how infection spreads through a community. We say a set of vertices in a
graph percolates if, when this set of vertices start off as infected, the whole
graph ends up infected. We study minimal percolating sets, that is, percolating
sets with no proper percolating subsets. In particular, we investigate
if there is any relation between the smallest and the largest minimal percolating
sets in bounded degree graph sequences.
A tournament is a complete graph where every edge has been given
an orientation. We look at the maximum number of directed k-cycles a
tournament can have and investigate when there exist tournaments with
many more k-cycles than expected in a random tournament.

##### Authors

Day, Alan Nicholas##### Collections

- Theses [2587]