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 …

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 …

On the Turán number of the hypercube Page 1 Forum of Mathematics, Sigma (2024), Vol. 12:e38
1–19 doi:10.1017/fms.2024.27 RESEARCH ARTICLE On the Turán number of the hypercube …

We show that for every integer m≥ 2 m ≥ 2 and large nn, every properly edge‐colored
graph on nn vertices with at least n (log n) 53 n\left (\log n\right)^ 53 edges contains a …

In this survey we aim to give a comprehensive overview of results using sublinear
expanders. The term sublinear expanders refers to a variety of definitions of expanders …

The Turán density of an-uniform hypergraph, denoted, is the limit of the maximum density of
an-vertex-uniform hypergraph not containing a copy of, as. Denote by the-uniform tight cycle …

We show that every $ r $-uniform hypergraph on $ n $ vertices which does not contain a tight
cycle has at most $ O (n^{r-1}(\log n)^ 5) $ edges. This is an improvement on the previously …

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 …