Maximising the number of induced cycles in a graph

We determine the maximum number of induced cycles that can be contained in a graph on $n\ge n_0$ vertices, and show that there is a unique graph that achieves this maximum. This answers a question of Tuza. We also determine the maximum number of odd or even cycles that can be contained in a graph on $n\ge n_0$ vertices and characterise the extremal graphs. This resolves a conjecture of Chv\'atal and Tuza from 1988.Comment: 36 page

Hypergraph Lagrangians I: the Frankl-F\"uredi conjecture is false

An old and well-known conjecture of Frankl and F\"{u}redi states that the Lagrangian of an $r$-uniform hypergraph with $m$ edges is maximised by an initial segment of colex. In this paper we disprove this conjecture by finding an infinite family of counterexamples for all $r \ge 4$. We also show that, for sufficiently large $t \in \mathbb{N}$, the conjecture is true in the range $\binom{t}{r} \le m \le \binom{t+1}{r} - \binom{t-1}{r-2}$.Comment: We split our original paper (arXiv:1807.00793v2) into two parts. This first part consists of 24 pages, including a one-page appendix. The second part appears in a new submission (arXiv:1907.09797

Bounding the Number of Hyperedges in Friendship rr-Hypergraphs

For $r \ge 2$, an $r$-uniform hypergraph is called a friendship $r$-hypergraph if every set $R$ of $r$ vertices has a unique 'friend' - that is, there exists a unique vertex $x \notin R$ with the property that for each subset $A \subseteq R$ of size $r-1$, the set $A \cup \{x\}$ is a hyperedge. We show that for $r \geq 3$, the number of hyperedges in a friendship $r$-hypergraph is at least $\frac{r+1}{r} \binom{n-1}{r-1}$, and we characterise those hypergraphs which achieve this bound. This generalises a result given by Li and van Rees in the case when $r = 3$. We also obtain a new upper bound on the number of hyperedges in a friendship $r$-hypergraph, which improves on a known bound given by Li, van Rees, Seo and Singhi when $r=3$.Comment: 14 page

On Saturated kk-Sperner Systems

Given a set $X$, a collection $\mathcal{F}\subseteq\mathcal{P}(X)$ is said to be $k$-Sperner if it does not contain a chain of length $k+1$ under set inclusion and it is saturated if it is maximal with respect to this property. Gerbner et al. conjectured that, if $|X|$ is sufficiently large with respect to $k$, then the minimum size of a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ is $2^{k-1}$. We disprove this conjecture by showing that there exists $\varepsilon>0$ such that for every $k$ and $|X| \geq n_0(k)$ there exists a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ with cardinality at most $2^{(1-\varepsilon)k}$. A collection $\mathcal{F}\subseteq \mathcal{P}(X)$ is said to be an oversaturated $k$-Sperner system if, for every $S\in\mathcal{P}(X)\setminus\mathcal{F}$, $\mathcal{F}\cup\{S\}$ contains more chains of length $k+1$ than $\mathcal{F}$. Gerbner et al. proved that, if $|X|\geq k$, then the smallest such collection contains between $2^{k/2-1}$ and $O\left(\frac{\log{k}}{k}2^k\right)$ elements. We show that if $|X|\geq k^2+k$, then the lower bound is best possible, up to a polynomial factor.Comment: 17 page

Saturation in the Hypercube and Bootstrap Percolation

Let $Q_d$ denote the hypercube of dimension $d$. Given $d\geq m$, a spanning subgraph $G$ of $Q_d$ is said to be $(Q_d,Q_m)$-saturated if it does not contain $Q_m$ as a subgraph but adding any edge of $E(Q_d)\setminus E(G)$ creates a copy of $Q_m$ in $G$. Answering a question of Johnson and Pinto, we show that for every fixed $m\geq2$ the minimum number of edges in a $(Q_d,Q_m)$-saturated graph is $\Theta(2^d)$. We also study weak saturation, which is a form of bootstrap percolation. A spanning subgraph of $Q_d$ is said to be weakly $(Q_d,Q_m)$-saturated if the edges of $E(Q_d)\setminus E(G)$ can be added to $G$ one at a time so that each added edge creates a new copy of $Q_m$. Answering another question of Johnson and Pinto, we determine the minimum number of edges in a weakly $(Q_d,Q_m)$-saturated graph for all $d\geq m\geq1$. More generally, we determine the minimum number of edges in a subgraph of the $d$-dimensional grid $P_k^d$ which is weakly saturated with respect to `axis aligned' copies of a smaller grid $P_r^m$. We also study weak saturation of cycles in the grid.Comment: 21 pages, 2 figures. To appear in Combinatorics, Probability and Computin

Characterising biochemical changes to hepatocellular carcinoma (HepG2) cells upon exposure to green tea extract using untargeted metabolomics

Complementary and alternative medicines (CAMs) have become the preferred medicine for many in replacement of conventional medicines due to cultural or financial reasons. Herbal CAMS, in particular, have become a popular choice of medicine for their purported health benefits. Green tea extract (GTE) contains the major catechins epigallocatechin-3-gallate, epicatechin gallate, epigallocatechin and epicatechin, all of which vary across different GTE products and have become the focus on research into its purported health benefits. However, there have been cases of GTE-induced hepatotoxicity, for which the biochemical pathways have not been characterised. This study elucidates compounds similarities and changes in catechin levels within several different GTE products, and biochemical pathways related to reactive oxygen species (ROS) production affected by acute GTE supplementation in an in vitro setting using metabolomic techniques. It was found that GTE hepatotoxicity significantly decreased amino acids, oxoacids and carboxylic acids at 1 mg/mL exposure but produced a different metabolite profile upon 0.1 mg/mL exposure. The results demonstrate that GTE hepatotoxicity is a dose-dependent process that induces ROS production, ATP depletion and apoptosis, which corroborates prior knowledge on this topic. These results utilise a novel field of research, metabolomics, to add insight into the biochemical mechanisms of GTE hepatotoxicity and to observe the mass spectral pattern and levels of four catechins in different GTE products: (+)-catechin, (-)-epicatechin, (-)-epigallocatechin and (-)-epigallocatechin-3-gallate. This will allow consumers to become more aware of herb-induced liver injury and provide data to aid the regulation of herbal CAMs

Extremal bounds for bootstrap percolation in the hypercube

The r-neighbour bootstrap percolation process on a graph G starts with an initial set A0 of “infected” vertices and, at each step of the process, a healthy vertex becomes infected if it has at least r infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of G eventually becomes infected, then we say that A0 percolates. We prove a conjecture of Balogh and Bollob ́as which says that, for fixed r and d →∞ , every percolating set in the d -dimensional hypercube has cardinality at least 1+ o (1) / r ( d r − 1 ). We also prove an analogous result for multidimensional rectangular grids. Our proofs exploit a connection between bootstrap percolation and a related process, known as weak saturation. In addition, we improve on the best known upper bound for the minimum size of a percolating set in the hypercube. In particular, when r = 3, we prove that the minimum cardinality of a percolating set in the d -dimensional hypercube is ⌈ d (d +3) / 6 ⌉ + 1 for all d ≥ 3