The problem of finding a maximal dense subgraph of a power-law random graph $G\left(n,\alpha \right)$ is considered for every value of density $c\in \left(0,1\right)$ and for every $\alpha \in \left(0,+\infty \right)$. It is shown that in case $\alpha <2$ a maximal $c$-dense subgraph has size ${\mathrm{\Theta }}_p\left(n^{1-\alpha ∕2}\right)$, in case $\alpha =2$ it is limited whp, and in case $\alpha >2$ it is whp less than $\frac{4}{c}$.

For nonnegative integers $k\le n$, we prove a combinatorial identity for the $p$-binomial coefficient $\left[b\right]{\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)}_p$ based on abelian $p$-groups. A purely combinatorial proof of this identity is not known. While proving this identity, for $r\in \mathbb{N}\cup \left\{0\right\},s\in \mathbb{N}$ and $p$ a prime, we present a purely combinatorial formula for the number of subgroups of ${\mathbb{Z}}^s$ of finite index $p^r$ with quotient isomorphic to the finite abelian $p$-group of type $\underset{¯}{\lambda }$ , which is a partition of $r$ into at most $s$ parts. This purely combinatorial formula is similar to that for the enumeration of subgroups of a certain type in a finite abelian $p$-group obtained by Lynne Marie Butler. As consequences, this combinatorial formula gives rise to many enumeration formulae that involve polynomials in $p$ with nonnegative integer coefficients.

Motivated by long-standing conjectures on the discretization of classical inequalities in the geometry of numbers, we investigate a new set of parameters, which we call packing minima, associated to a convex body $K$ and a lattice $\mathrm{\Lambda }$. These numbers interpolate between the successive minima of $K$ and the inverse of the successive minima of the polar body of $K$ and can be understood as packing counterparts to the covering minima of Kannan & Lovász (1988).

As our main results, we prove sharp inequalities that relate the volume and the number of lattice points in $K$ to the sequence of packing minima. Moreover, we extend classical transference bounds and discuss a natural class of examples in detail.

Let $X$ be an $n$-element set, where $n$ is even. We refute a conjecture of J. Gordon and Y. Teplitskaya, according to which, for every maximal intersecting family $\mathcal{ℱ}$ of $\frac{n}{2}$-element subsets of $X$, one can partition $X$ into $\frac{n}{2}$ disjoint pairs in such a way that no matter how we pick one element from each of the first $\frac{n}{2}-1$ pairs, the set formed by them can always be completed to a member of $\mathcal{ℱ}$ by adding an element of the last pair.

The above problem is related to classical questions in extremal set theory. For any $t\ge 2$, we call a family of sets $\mathcal{ℱ}\subset 2^X$ $t$-separable if there is a $t$-element subset $T\subseteq X$ such that for every ordered pair of elements $\left(x,y\right)$ of $T$, there exists $F\in \mathcal{ℱ}$ such that $F\cap \left\{x,y\right\}=\left\{x\right\}$. For a fixed $t$, $2\le t\le 5$, and $n\to \infty$, we establish asymptotically tight estimates for the smallest integer $s=s\left(n,t\right)$ such that every family $\mathcal{ℱ}$ with $\left|\mathcal{ℱ}\right|\ge s$ is $t$-separable.

Modularity is designed to measure the strength of a division of a network into clusters. For $n\in \mathbb{N}$, consider a set $S$ of $n$ elements. Let $V$ be the set of all subsets of $S$ of size $r$. Consider the graph $G\left(n,r,s\right)$ with vertices $V$ and edges that connect two vertices if and only if their intersection has size $s$. In this article, we find the exact modularity of $G\left(n,2,1\right)$ for $n\ge 5$.

The modularity of a graph is a value that shows how well the graph can be split into clusters. It is a key part in many clustering algorithms. In this paper, we improve the lower bound on the modularity of Johnson graphs significantly.