We show that if (Ω, Σ, μ) and (Ω′, Σ′, μ′) are probability spaces, then every regular operator T : Lp(μ) → Lq(μ′), 1 < p < ∞, 1 ≤ q < ∞, is thin if and only if it is strictly singular. We also show that if 0 ≤ S ≤ T : Lp(μ) → Lq(μ′), then T thin implies S is thin. We extend these results to some Köthe function spaces.

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.

Bollobás and Riordan introduce a Tutte polynomial for coloured graphs and matroids in [3]. We observe that this polynomial has an expansion as a sum indexed by the subsets of the ground-set of a coloured matroid, generalizing the subset expansion of the Tutte polynomial. We also discuss similar expansions of other contraction–deletion invariants of graphs and matroids.