Abstract
Starting with Willem Mantel in 1907 and continuing with the work of Turan and Erd os in the midtwentieth century, the extremal properties of graphs (or hypergraphs) avoiding a fixed subgraph (or subhypergraph) have been extensively studied, up to the present day. This is the field of socalled ‘Turan problems’; it has had much impact on other areas of Mathematics, such as number theory and geometry, as well as on combinatorics and theoretical computer science.The classical ‘Turan problem’ for a fixed runiform hypergraph F is the following: for
each positive integer n, what is the maximum number ex(n, F ) of edges we may take in a r
uniform hypergraph H on n vertices that contains no copy of F? The limit of ex(n, F )/(n choose r)
as n tends to infinity is called the Turan density of F, and is usually denoted by π(F).
In this thesis we study a natural and important class of Tur ́an problems for hypergraphs, posed by Bollobas, Leader and Malvenuto, and independently by Johnson and Talbot and (again independently) by Bukh. For integers r ≥ 3 and t ≥ 2, we define an runiform tdaisy Drt and consider the Turan problem for F = Drt. The exact value of π(Drt) is not known for any t ≥ 2,r ≥ 3. It was conjectured by Bollobas, Leader and Malvenuto in [3] (and independently by Bukh; an equivalent conjecture was made independently by Johnson and Talbot) that the Turan densities of tdaisies satisfy lim π(Drt ) = 0 for all t ≥ 2; this is still open for r→∞ and all values of t. We give lower bounds for the Turan densities of runiform tdaisies, im
proving the best known lower bound from exponential to polynomial in r.
Date of Award  27 Sept 2022 

Original language  English 
Awarding Institution 

Supervisor  David C Ellis (Supervisor) 