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 …

In the 1960's, Erdős and Gallai conjectured that the edges of any n-vertex graph can be
decomposed into O (n) cycles and edges. We improve upon the previous best bound of O (n …

Abstract A graph G=(V, E) is called an expander if every vertex subset U of size up to| V|/2
has an external neighborhood whose size is comparable to| U|. Expanders have been a …

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 …

We develop a general embedding method based on the Friedman-Pippenger tree
embedding technique and its algorithmic version, enhanced with a roll-back idea allowing a …

Given any integers s, t⩾ 2, we show that there exists some c= c (s, t)> 0 such that any K s, t‐
free graph with average degree d contains a subdivision of a clique with at least cds/2 (s− 1) …

What is the maximum number of copies of a fixed forest T in an n-vertex graph in a graph
class as? We answer this question for a variety of sparse graph classes. In particular, we …

We show that for any integer t≥ 2, every properly edge-coloured graph on n vertices with
more than n 1+ o (1) edges contains a rainbow subdivision of K t. Note that this bound on the …

Let H be a planar graph. By a classical result of Robertson and Seymour, there is a function
f: ℕ→ ℝ such that for all k∊ ℕ and all graphs G, either G contains k vertex-disjoint subgraphs …

Komlós conjectured in 1981 that among all graphs with minimum degree at least d, the
complete graph K d+ 1 minimises the number of Hamiltonian subsets, where a subset of …