In this paper we consider a bipartite version of Schütte's well-known tournament problem. A bipartite tournament $T=(A,B,E)$ with teams $A$ and $B$, and set of arcs $E$, has the property $S_{k,l}$ if for any subsets $K\subseteq A$ and $L\subseteq B$, with $|K| =k$ and $| L | =l$, there exist conquerors of $K$ and $L$ in opposite teams. The task is to estimate, for fixed $k$ and $l$, the minimum number $f(k,l)=| A | + | B | $ of players in a tournament satisfying property $S_{k,l}$. We achieve this goal by reformulating the problem in terms of intersecting set families and applying probabilistic as well as constructive methods. Intriguing connections with some famous problems of this area have emerged in this way, leading to new open questions.

In this article we introduce combinatorial multicolour discrepancies and generalize several classical results from $2$-colour discrepancy theory to $c$ colours ($c \geq 2$). We give a recursive method that constructs $c$-colourings from approximations of $2$-colour discrepancies. This method works for a large class of theorems, such as the ‘six standard deviations’ theorem of Spencer (1985), the Beck–Fiala (1981) theorem, the results of Matoušek, Wernisch and Welzl (1994) and Matoušek (1995) for bounded VC-dimension, and Matoušek and Spencer's (1996) upper bound for the arithmetic progressions. In particular, the $c$-colour discrepancy of an arbitrary hypergraph ($n$ vertices, $m$ hyperedges) is \[ \OO\Bigl(\sqrt{\tfrac n c\,\log m}\Bigr). \] If $m = \OO(n)$, then this bound improves to \[ \OO\Bigl(\sqrt{\tfrac n c\,\log c}\Bigr). \]

On the other hand there are examples showing that discrepancy in $c$ colours can not be bounded in terms of two-colour discrepancies in general, even if $c$ is a power of 2. For the linear discrepancy version of the Beck–Fiala theorem, the recursive approach also fails.

Here we extend the method of floating colours via tensor products of matrices to multicolourings, and prove multicolour versions of the Beck–Fiala theorem and the Bárány–Grinberg theorem. Using properties of the tensor product we derive a lower bound for the $c$-colour discrepancy of general hypergraphs. For the hypergraph of arithmetic progressions in $\{1, \ldots, n\}$ this yields a lower bound of $\frac{1}{25 \sqrt c} \sqrt[4]{n}$ for the discrepancy in $c$ colours. The recursive method shows an upper bound of $\OO(c^{-0.16} \sqrt[4]{n})$

For a random graph on n vertices where the edges appear with individual rates, we give exact formulas for the expected time at which the number of components has gone down to k and the expected length of the corresponding minimal spanning forest.

For a random bipartite graph we give a formula for the expected time at which a k-assignment appears. This result has a bearing on the random assignment problem.

Let $G$ be a cyclic group of order $n$ and let $\mu = \{x_1,x_2, \dots, x_m\}$ be a sequence of elements of $G$. Let $k$ be the number of distinct values taken by the sequence $\mu$. Let $n\wedge \mu$ be the set of the $n$-subsequence sums.

We show that one of the following conditions holds:

1. $\mu$ has a value repeated $n-k+3$ times

2. $n\wedge \mu$ contains a non-null subgroup

3. $|n\wedge \mu|\geq m-n+k-2.$

We conjecture that the last condition could be improved to $|n\wedge \mu|\geq m-n+k-1$. This conjecture generalizes several known results. We also obtain a generalization of a recent result due to Bollobás and Leader.

A set $S\subset \R^d$ is $C$-Lipschitz in the$x_i$-coordinate, where $C>0$ is a real number, if, for every two points $a,b\in S$, we have $|a_i-b_i|\leq C \max\{|a_j-b_j|\sep j=1,2,\ldots,d,\,j\neq i\}$. Motivated by a problem of Laczkovich, the author asked whether every $n$-point set in $\Rd$ contains a subset of size at least $cn^{1-1/d}$ that is $C$-Lipschitz in one of the coordinates, for suitable constants $C$ and $c>0$ (depending on $d$). This was answered negatively by Alberti, Csörnyei and Preiss. Here it is observed that a combinatorial result of Ruzsa and Szemerédi implies the existence of a 2-Lipschitz subset of size $n^{1/2}\varphi(n)$ in every $n$-point set in $\R^3$, where $\varphi(n)\to\infty$ as $n\to\infty$.

This paper is concerned with path techniques for quantitative analysis of the logarithmic Sobolev constant on a countable set. We present new upper bounds on the logarithmic Sobolev constant, which generalize those given by Sinclair [20], in the case of the spectral gap constant involving path combinatorics. Some examples of applications are given. Then, we compare our bounds to the Hardy constant in the particular case of birth and death processes. Finally, following the approach of Rosenthal in [18], we generalize our bounds to continuous sets.

Let G be a simple 3-connected graph with at least five vertices. Tutte [13] showed that G has at least one contractible edge. Thomassen [11] gave a simple proof of this fact and showed that contractible edges have many applications. In this paper, we show that there are at most $\frac{|V(G)|}{5}$ vertices that are not incident to contractible edges in a 3-connected graph G. This bound is best-possible. We also show that if a vertex v is not incident to any contractible edge in G, then v has at least four neighbours having degree three, and each such neighbour is incident to exactly two contractible edges. We give short proofs of several results on contractible edges in 3-connected graphs as well. We also study the contractible elements for k-connected matroids. We partially solve an open problem for regular matroids.