A family of sets is intersecting if no two of its members are disjoint, and has the Erdős–Ko–Rado property (or is EKR) if each of its largest intersecting subfamilies has non-empty intersection.

Denote by ${{\cal H}_k}(n,p)$ the random family in which each k-subset of {1, …, n} is present with probability p, independent of other choices. A question first studied by Balogh, Bohman and Mubayi asks:

\begin{equation} {\rm{For what }}p = p(n,k){\rm{is}}{{\cal H}_k}(n,p){\rm{likely to be EKR}}? \end{equation}

Here, for fixed c < 1/4, and $k \lt \sqrt {cn\log n} $ we give a precise answer to this question, characterizing those sequences p = p(n, k) for which

\begin{equation} {\mathbb{P}}({{\cal H}_k}(n,p){\rm{is EKR}}{\kern 1pt} ) \to 1{\rm{as }}n \to \infty . \end{equation}

Denote by ${\mathcal H}_k$(n, p) the random k-graph in which each k-subset of {1,. . .,n} is present with probability p, independent of other choices. More or less answering a question of Balogh, Bohman and Mubayi, we show: there is a fixed ε > 0 such that if n = 2k + 1 and p > 1 - ε, then w.h.p. (that is, with probability tending to 1 as k → ∞), ${\mathcal H}_k$(n, p) has the ‘Erdős–Ko–Rado property’. We also mention a similar random version of Sperner's theorem.

In this paper we compute the absorbing time Tn of an n-dimensional discrete-time Markov chain comprising n components, each with an absorbing state and evolving in mutual exclusion. We show that the random absorbing time Tn is well approximated by a deterministic time tn that is the first time when a fluid approximation of the chain approaches the absorbing state at a distance 1 / n. We provide an asymptotic expansion of tn that uses the spectral decomposition of the kernel of the chain as well as the asymptotic distribution of Tn, relying on extreme values theory. We show the applicability of this approach with three different problems: the coupon collector, the erasure channel lifetime, and the coupling times of random walks in high-dimensional spaces.

We study sum-free sets in sparse random subsets of even-order abelian groups. In particular, we determine the sharp threshold for the following property: the largest such set is contained in some maximum-size sum-free subset of the group. This theorem extends recent work of Balogh, Morris and Samotij, who resolved the case G = ℤ2n, and who obtained a weaker threshold (up to a constant factor) in general.

Let G be a triangle-free graph with n vertices and average degree t. We show that G contains at least

${\exp\biggl({1-n^{-1/12})\frac{1}{2}\frac{n}{t}\ln t} \biggl(\frac{1}{2}\ln t-1\biggr)\biggr)}$

${e^{(c_1-o(1)) \sqrt{n} \ln n}}$

$c_1 = \sqrt{\ln 2}/4 = 0.208138 \ldots$

${e^{(c_2+o(1))\sqrt{n}\ln n}}$

$c_2 = 2\sqrt{\ln 2} = 1.665109 \ldots$

Let H be a (k+1)-uniform linear hypergraph with n vertices and average degree t. We also show that there exists a constant ck such that the number of independent sets in H is at least

${\exp\biggl({c_{k} \frac{n}{t^{1/k}}\ln^{1+1/k}{t}\biggr})}.$

$\Omega\biggl(\frac{n}{t^{1/k}} \ln^{1/k}t\biggr).$

Given a set $A\subset\mathbb{Z}_{N}$, we say that a function $f\colon A \to \mathbb{Z}_{N}$ is a Freiman homomorphism if f(a)+f(b)=f(c)+f(d) whenever a,b,c,d ∈ A satisfy a+b=c+d. This notion was introduced by Freiman in the 1970s, and plays an important role in the field of additive combinatorics. We say that A is linear if the only Freiman homomorphisms are functions of the form f(x) = ax+b.

Suppose the elements of A are chosen independently at random, each with probability p. We shall look at the following question: For which values of p=p(N) is A linear with high probability as N → ∞? We show that if p=(2logN − ω(N))1/3N−2/3, where ω(N) → ∞ as N → ∞, then A is not linear with high probability, whereas if p=N−1/2+ε for any ε>0 then A is linear with high probability.