We present several results in extremal graph and hypergraph theory of topological nature.
First, we show that if $\alpha> 0$ and $\ell=\Omega (\frac {1}{\alpha}\log\frac {1}{\alpha}) $ is …

An edge-coloured graph is said to be rainbow if no colour appears more than once.
Extremal problems involving rainbow objects have been a focus of much research over the …

In 1975, Erd̋s asked the following question: What is the maximum number of edges that an-
vertex graph can have without containing a cycle with all diagonals? Erd̋s observed that …

We show that there is a constant cc such that any 3‐uniform hypergraph HH with nn vertices
and at least cn 5/2 cn^5/2 edges contains a triangulation of the real projective plane as a …

Our first main result is a uniform bound, in every dimension $ k\in\mathbb N $, on the
topological Tur\'an numbers of $ k $-dimensional simplicial complexes: for each …

We prove that the maximum number of edges in a 3-uniform linear hypergraph on vertices
containing no 2-regular subhypergraph is. This resolves a conjecture of Dellamonica …

We study a higher-dimensional analogue of the {Random Travelling Salesman Problem}: let
the complete $ d $-dimensional simplicial complex $ K_n^{d} $ on $ n $ vertices be …

We consider the hypergraph Tur\'an problem of determining $\mathrm {ex}(n, S^ d) $, the
maximum number of facets in a $ d $-dimensional simplicial complex on $ n $ vertices that …

The topological Tur\'an number $\mathrm {ex} _ {\hom}(n, X) $ of a 2-dimensional simplicial
complex $ X $ asks for the maximum number of edges in an $ n $-vertex 3-uniform …

This thesis consists of four parts, each on a different problem in extremal or probabilistic
combinatorics. Chapters 2 and 3 center around hypergraph versions of foundational …