Let graph $G=(V,E)$ and integer $b\geq 1$ be given. A set $S\subseteq V$ is said to be $b$-independent if $u,v\in S$ implies $d_G(u,v)>b$, where $d_G(u,v)$ is the shortest distance between $u$ and $v$ in $G$. The $b$-independence number $\a_b(G)$ is the size of the largest $b$-independent subset of $G$. When $b=1$ this reduces to the standard definition of independence number.

We study this parameter in relation to the random graph $G_{n,p},\,p=d/n$, in particular, when $d$ is a large constant. We show that w.h.p. if $d\geq d_{\epsilon, b}$, $$ \left| \alpha_b(G_{n,p}) - \frac{2bn}{d^b} \biggl(\log{d} - \frac{\log{\log{d}}}{b} - \frac{\log{2b}}{b} + \frac{1}{b}\biggr)\right| \leq \frac{\epsilon n}{d^b}.$$

Improving an old result of Clarkson, Edelsbrunner, Guibas, Sharir and Welzl, we show that the number of distinct distances determined by a set $P$ of $n$ points in three-dimensional space is $\Omega(n^{77/141-\varepsilon})=\Omega(n^{0.546})$, for any $\varepsilon>0$. Moreover, there always exists a point $p\in P$ from which there are at least so many distinct distances to the remaining elements of $P$. The same result holds for points on the three-dimensional sphere. As a consequence, we obtain analogous results in higher dimensions.

We show that for every $\varepsilon\,{>}\,0$ there exists an $r_0\,{=}\,r_0(\varepsilon)$ such that, for all integers $r\,{\ge}\, r_0$, every graph of average degree at least $r+\varepsilon$ and girth at least 1000 contains a subdivision of $K_{r+2}$. Combined with a result of Mader this implies that, for every $\varepsilon\,{>}\,0$, there exists an $f(\varepsilon)$ such that, for all $r\,{\ge}\, 2$, every graph of average degree at least $r+\varepsilon$ and girth at least $f(\varepsilon)$ contains a subdivision of $K_{r+2}$. We also prove a more general result concerning subdivisions of arbitrary graphs.

Using representation theory, we obtain a necessary and sufficient condition for a discrete-time Markov chain on a finite state space $E$ to be representable as $$\Psi_n \Psi_{n-1} \cdots \Psi_1 z,\quad n \geq 0,$$ for any $z \in E$, where the $\Psi_i$ are independent, identically distributed random permutations taking values in some given transitive group of permutations on $E$. The condition is particularly simple when the group is 2-transitive on $E$. We also work out the explicit form of our condition for the dihedral group of symmetries of a regular polygon.

What simple condition on a graph $G$ will ensure that $G\,{\succ}\,K_t$? As usual, $G\,{\succ}\,K_t$ means that $K_t$ is a minor of the graph $G$ (in other words, $G$ has vertex disjoint connected subgraphs $W_1,\ldots,W_t$ and at least one edge between $W_i$ and $W_j$, $1\,{\le}\, i<j\,{\le}\, t$).

We compute the fat-shattering function and the level fat-shattering function for important classes of affine functions. We observe that the level fat-shattering function and the fat-shattering function are identical for these classes. In addition we observe that the notion that adding the constant term to linear functions increases the dimension by at most 1 is incorrect for fat-shattering and level fat-shattering.

We show that all maximum length paths in a connected circular arc graph have non-empty intersection.

In this paper we prove several point selection theorems concerning objects ‘spanned’ by a finite set of points. For example, we show that for any set $P$ of $n$ points in $\R^2$ and any set $C$ of $m \,{\geq}\, 4n$ distinct pseudo-circles, each passing through a distinct pair of points of $P$, there is a point in $P$ that is covered by (i.e., lies in the interior of) $\Omega(m^2/n^2)$ pseudo-circles of $C$. Similar problems involving point sets in higher dimensions are also studied.

Most of our bounds are asymptotically tight, and they improve and generalize results of Chazelle, Edelsbrunner, Guibas, Hershberger, Seidel and Sharir [8], where weaker bounds for some of these cases were obtained.

We give results on the strong connectivity for spaces of sparse random digraphs specified by degree sequence. A full characterization is provided, in probability, of the fan-in and fan-out of all vertices including the number of vertices with small ($o(n)$) and large ($cn$) fan-in or fan-out. We also give the size of the giant strongly connected component, if any, and the structure of the bow-tie digraph induced by the vertices with large fan-in or fan-out. Our results follow a direct analogy of the extinction probabilities of classical branching processes.

We look at a model of random graphs suggested by Gilbert: given an integer $n$ and $\delta > 0$, scatter $n$ vertices independently and uniformly on a metric space, and then add edges connecting pairs of vertices of distance less than $\delta$ apart.

We consider the asymptotics when the metric space is the interval [0, 1], and $\delta = \delta(n)$ is a function of $n$, for $n \to \infty$. We prove that every upwards closed property of (ordered) graphs has at least a weak threshold in this model on this metric space. (But we do find a metric space on which some upwards closed properties do not even have weak thresholds in this model.) We also prove that every upwards closed property with a threshold much above connectivity's threshold has a strong threshold. (But we also find a sequence of upwards closed properties with lower thresholds that are strictly weak.)

How ‘tightly’ can we pack a given number of $r$-sets of an $n$-set? To be a little more precise, let $X=[n]=\{ 1,\ldots,n \}$, and let $X^r=\{ A\subset X : |A|=r \}$. For a set system $\mathcal{A}\subset X^r $, the neighbourhood of $\mathcal{A}$ is $N(\mathcal{A})=\{ B \in X^r: |B \bigtriangleup A|\le 2 \hbox{ for some }A \in \mathcal{A} \}$. In other words, $N(\mathcal{A})$ consists of those $r$-sets that are either in $\mathcal{A}$ or are ‘adjacent’ to it, in the sense that they are at minimal Hamming distance (i.e., distance 2) from some point of it. Given $|\mathcal{A}|$, how small can $|N(\mathcal{A})|$ be?

Sampling formulas describe probability laws of exchangeable combinatorial structures like partitions and compositions. We give a brief account of two known parametric families of sampling formulas for compositions and add a new family to the list.

We give a quantitative proof that, for sufficiently large $N$, every subset of $[N]^2$ of size at least $\delta N^2$ contains a square, i.e., four points with coordinates $\{(a,b),(a+d,b),(a,b+d),(a+d,b+d)\}$.

Baranyai's partition theorem states that the edges of the complete $r$-graph on $n$ vertices can be partitioned into $1$-factors provided that $r$ divides $n$. Fon-der-Flaass has conjectured that for $r=3$ such a partitioning exists with the property that any two $1$-factors are ‘far apart’ in some natural sense.

Our aim in this note is to prove that the Fon-der-Flaass conjecture is not always true: it fails for $n=12$. Our methods are based on some new ‘auxiliary’ hypergraphs.

Given a set $L$ of $n$ lines in ${\mathbb R}^3$, joints are points in ${\mathbb R}^3$ that are incident to at least three non-coplanar lines in $L$. We show that there are at most $O(n^{5/3})$ incidences between $L$ and the set of its joints.

This result leads to related questions about incidences between $L$ and a set $P$ of $m$ points in ${\mathbb R}^3$. First, we associate with every point $p \in P$ the minimum number of planes it takes to cover all lines incident to $p$. Then the sum of these numbers is at most \[ O\big(m^{4/7}n^{5/7}+m+n\big).\] Second, if each line forms a fixed given non-zero angle with the $xy$-plane – we say the lines are equally inclined – then the number of (real) incidences is at most \[ O\big(\min\big\{m^{3/4}n^{1/2}\kappa(m),\ m^{4/7}n^{5/7}\big\} + m + n\big) , \] where $\kappa(m) \,{=}\, (\log m)^{O(\alpha^2(m))}$, and $\alpha(m)$ is the slowly growing inverse Ackermann function. These bounds are smaller than the tight Szemerédi–Trotter bound for point–line incidences in $\reals^2$, unless both bounds are linear. They are the first results of this type on incidences between points and $1$-dimensional objects in $\reals^3$. This research was stimulated by a question raised by G. Elekes.

I show that the zeros of the chromatic polynomials $P_G(q)$ for the generalized theta graphs $\Theta^{(s,p)}$ are, taken together, dense in the whole complex plane with the possible exception of the disc $|q-1| < 1$. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) $Z_G(q,v)$ outside the disc $|q+v| < |v|$. An immediate corollary is that the chromatic roots of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha–Kahane–Weiss theorem on the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof.