This paper is an extended version of the lecture given by the second author delivered at the conference ‘Combinatorics: Walter Deuber Memorial Meeting’ held on 7–8 October 2002 at Humboldt-Universität zu Berlin. Regretfully, this topic was to be the last that fascinated Walter Deuber's mind. We wrote this article in remembrance of that.

Let ${\cal T}(n,m)$ denote the set of all labelled triangle-free graphs with $n$ vertices and exactly $m$ edges. In this paper we give a short self-contained proof of the fact that there exists a constant $C>0$ such that, for all $m\geq Cn^{3/2}\sqrt{\log n}$, a graph chosen uniformly at random from ${\cal T}(n,m)$ is with probability $1-o(1)$ bipartite.

We consider several extremal problems concerning representations of graphs as distance graphs on the integers. Given a graph $G=(V,E)$, we wish to find an injective function $\phi:V\to{\mathbb Z}^+=\{1,2,\dots\}$ and a set ${\mathcal D}\subset{\mathbb Z}^+$ such that $\{u,v\}\in E$ if and only if $|\phi(u)-\phi(v)|\in{\mathcal D}$.

Let $s(n)$ be the smallest $N$ such that any graph $G$ on $n$ vertices admits a representation $(\phi_G,{\mathcal D}_G)$ such that $\phi_G(v)\leq N$ for all $v\in V(G)$. We show that $s(n)=(1+o(1))n^2$ as $n\to\infty$. In fact, if we let $s_r(n)$ be the smallest $N$ such that any $r$-regular graph $G$ on $n$ vertices admits a representation $(\phi_G,{\mathcal D}_G)$ such that $\phi_G(v)\leq N$ for all $v\in V(G)$, then $s_r(n)=(1+o(1))n^2$ as $n\to\infty$ for any $r=r(n)\gg\log n$ with $rn$ even for all $n$.

Given a graph $G=(V,E)$, let $D_{\rm e}(G)$ be the smallest possible cardinality of a set ${\mathcal D}$ for which there is some $\phi\:V\to{\mathbb Z}^+$ so that $(\phi,{\mathcal D})$ represents $G$. We show that, for almost all $n$-vertex graphs $G$, we have \begin{equation*} D_{\rm e}(G)\geq\frac{1}{2}\binom{n}{2}-(1+o(1))n^{3/2}(\log n)^{1/2}, \end{equation*} whereas for some $n$-vertex graph $G$, we have \begin{equation*} D_{\rm e}(G)\geq\binom{n}{2}-n^{3/2}(\log n)^{1/2+o(1)}.\end{equation*} Further extremal problems of similar nature are considered.

We present a programme of characterizing Ramsey classes of structures by a combination of the model theory and combinatorics. In particular, we relate the classification programme of countable homogeneous structures (of Lachlan and Cherlin) to the classification of Ramsey classes. As particular instances of this approach we characterize all Ramsey classes of graphs, tournaments and partial ordered sets. We fully characterize all monotone Ramsey classes of relational systems (of any type). We also carefully discuss the role of (admissible) orderings which lead to a new classification of Ramsey properties by means of classes of order-invariant objects.

The aim of this paper is to point to a difference between binary and hyperary structures. The modular counting functions of a class of structures defined by a sentence of second-order monadic logic with equality, based on binary relations, are ultimately periodic. However, this is not the case for sentences based on quaternary relations.

In this article we extend the results about Gorenstein modules and Foxby duality to a non-commutative setting. This is done in §3 of the paper, where we characterize the Auslander and Bass classes which arise whenever we have a dualizing module associated with a pair of rings. In this situation it is known that flat modules have finite projective dimension. Since this property of a ring is of interest in its own right, we devote §2 of the paper to a consideration of such rings. Finally, in the paper’s final section, we consider a natural generalization of the notions of Gorenstein modules which arises when we are in the situation of §3, i.e. when we have a dualizing module.

AMS 2000 Mathematics subject classification: Primary 16D20

Let $G$ be a noncomplete $k$-connected graph such that the graphs obtained from contracting any edge in $G$ are not $k$-connected, and let $t(G)$ denote the number of triangles in $G$. Thomassen proved $t(G) \geq 1$, which was later improved by Mader to $t(G) \geq \frac{1}{3}|V(G)|$.

Here we show $t(G) \geq \frac{2}{3}|V(G)|$ (which is best possible in general).

Furthermore it is proved that, for $k \geq 4$, a $k$-connected graph without two disjoint triangles must contain an edge not contained in a triangle whose contraction yields a $k$-connected graph. As an application, for $k \geq 4$ every $k$-connected graph $G$ admits two disjoint induced cycles $C_1,C_2$ such that $G-V(C_1)$ and $G-V(C_2)$ are $(k-3)$-connected.

A new asymptotic approach is suggested for studying spectra of linear differential operators with periodic coefficients. The resulting formal recurrent procedure and its rigorous justification allow us to prove a classical theorem on the density of states in one dimension.

AMS 2000 Mathematics subject classification: Primary 34L20. Secondary 74J99; 78M35

Let $\mathcal{A}$ be a closed, point-separating sub-algebra of $C_0(X)$, where $X$ is a locally compact Hausdorff space. Assume that $X$ is the maximal ideal space of $\mathcal{A}$. If $f\in\mathcal{A}$, the set $f(X)\cup\{0\}$ is denoted by $\sigma(f)$. After characterizing the points of the Choquet boundary as strong boundary points, we use this equivalence to provide a natural extension of the theorem in [10], which, in turn, was inspired by the main result in [6], by proving the ‘Main Theorem’: if $\varPhi:\mathcal{A}\rightarrow\mathcal{A}$ is a surjective map with the property that $\sigma(fg)=\sigma(\varPhi(f)\varPhi(g))$ for every pair of functions $f,g\in\mathcal{A}$, then there is an onto homeomorphism $\varLambda:X\rightarrow X$ and a signum function $\epsilon(x)$ on $X$ such that

$$ \varPhi(f)(\varLambda(x))=\epsilon(x)f(x) $$

for all $x\in X$ and $f\in\mathcal{A}$.

AMS 2000 Mathematics subject classification: Primary 46J10; 46J20

We consider algorithms for group testing problems when nothing is known in advance about the number of defectives. Du and Hwang suggested measuring the quality of such algorithms by its so-called (first) competitive ratio (see the Introduction). Later, Du and Park suggested a second kind of competitive ratio. For each kind of competitiveness, we improve the best-known bounds: in the first case, from 1.65 to $1.5+\ep$, and in the second from 16 to 4.

In this paper we extract some conclusions about Newton non-degenerate ideals and the computation of Łojasiewicz exponents relative to this kind of ideal. This motivates us to study the Newton non-degeneracy condition on the Jacobian ideal of a given analytic function germ $f:(\mathbb{C}^n,0)\to(\mathbb{C},0)$. In particular, we establish a connection between Newton non-degenerate functions and functions whose Jacobian ideal is Newton non-degenerate.

AMS 2000 Mathematics subject classification: Primary 32S05. Secondary 57R45

We study a nonlinear second-order periodic problem driven by the scalar $p$-Laplacian with a non-smooth potential. We consider the so-called doubly resonant situation allowing complete interaction (resonance) with both ends of the spectral interval. Using variational methods based on the non-smooth critical-point theory for locally Lipschitz functions and an abstract minimax principle concerning linking sets we establish the solvability of the problem.

AMS 2000 Mathematics subject classification: Primary 34B15; 34C25

In this paper we study the spaces of operator-Lipschitz functions and the spaces of functions closed to them: commutator bounded. Apart from the standard operator norm on $B(H)$, we consider a rich variety of symmetric operator norms and spaces of operator-Lipschitz functions with respect to these norms. Our approach is aimed at the investigation of the interrelation and hierarchy of these spaces and of the intrinsic properties of operator-Lipschitz functions.

AMS 2000 Mathematics subject classification: Primary 47A56. Secondary 47L20

This paper deals with a reaction–diffusion model with inner absorptions and coupled nonlinear boundary conditions of exponential type. The critical exponents are described via a pair of parameters that satisfy a certain matrix equation containing all the six nonlinear exponents of the system. Whether the solutions blow up or not is determined by the signs of the two parameters. A more precise analysis, depending on the geometry of $\varOmega$ and the absorption coefficients, is proposed for the critical sign of the parameters.

AMS 2000 Mathematics subject classification: Primary 35K55; 35B33

Let $L$ be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. Let $f:L\to\{\pm1\}$ be a homomorphism and, for $\alpha=\sum\alpha_\ell\ell\in\mathbb{Z}L$, define $\alpha^f=\sum f(\ell)\alpha_\ell\ell^{-1}$. Call $\alpha$ f-unitary if $\alpha^f=\alpha^{-1}$ or $\alpha^f=-\alpha^{-1}$. In this paper, we identify the RA loops $L$ with the property that all units in $\mathbb{Z}L$ are $f$-unitary. Along the way, we extend a famous theorem of Higman to a case still undecided in group rings.

AMS 2000 Mathematics subject classification: Primary 20N05. Secondary 17D05; 16S34; 16U60

The Ramsey Schur number $RS(s,t)$ is the smallest $n$ such that every 2-colouring of the edges of $K_n$ with vertices $1,2,\ldots,n$ contains a green $K_s$ or there are vertices $x_1,x_2,\ldots,x_t$ fulfilling the equation $x_1+x_2+\cdots+x_{t-1}=x_t$ and all edges $(x_i,x_j)$ are red. We prove $RS(3,3)=11, RS(3,t)=t^2-3$ for $t\equiv1\ (\mbox{mod}\ 6)$ and $t=8$, and $RS(3,t)\geq t^2-3$.

We consider elementary operators on centrally closed prime algebras that are Lie (or Jordan) homomorphisms or commutativity preservers.

AMS 2000 Mathematics subject classification: Primary 16N60. Secondary 16R50; 47B47

Complete disorder is impossible – this theme of Ramsey Theory, as stated by Theodore S. Motzkin, was a guiding theme throughout Walter Deuber's scientific life.

The singular boundary-value problem $(g(x'(t)))'=\mu f(t,x(t),x'(t))$, $x(0)=x(T)=0$ and $\max\{x(t):0\le t\le T\}=A$ is considered. Here $\mu$ is the parameter and the negative function $f(t,u,v)$ satisfying local Carathéodory conditions on $[0,T]\times(0,\infty)\times(\mathbb{R}\setminus\{0\})$ may be singular at the values $u=0$ and $v=0$ of the phase variables $u$ and $v$. The paper presents conditions which guarantee that for any $A>0$ there exists $\mu_A>0$ such that the above problem with $\mu=\mu_A$ has a positive solution on $(0,T)$. The proofs are based on the regularization and sequential techniques and use the Leray–Schauder degree and Vitali’s convergence theorem.

AMS 2000 Mathematics subject classification: Primary 34B16; 34B18