Properly colored subgraphs and rainbow subgraphs in edge‐colorings with local constraints

We consider a canonical Ramsey type problem. An edge‐coloring of a graph is called m‐good if each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let f(m, G) denote the smallest n such that every m‐good edge‐coloring of K_n yields a properly edge‐colored copy of G, and let g(m, G) denote the smallest n such that every m‐good edge‐coloring of K_n yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs G = K_t, we have c₁mt²/ln t ≤ f(m, K_t) ≤ c₂mt², and cmt³/ln t ≤ g(m, K_t) ≤ cmt³/ln t, where c₁, c₂, c, c are absolute constants. We also give bounds on f(m, G) and g(m, G) for general graphs G in terms of degrees in G. In particular, we show that for fixed m and d, and all sufficiently large n compared to m and d, f(m, G) = n for all graphs G with n vertices and maximum degree at most d. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2003     

The longest cycle of a graph with a large minimal degree

We show that every graph G on n vertices with minimal degree at least n/k contains a cycle of length at least [n/(k ? 1)]. This verifies a conjecture of Katchalski. When k = 2 our result reduces to the classical theorem of Dirac that asserts that if all degrees are at least 1/2n then G is Hamiltonian.     

On the maximum number of Hamiltonian paths in tournaments

By using the probabilistic method, we show that the maximum number of directed Hamiltonian paths in a complete directed graph with n vertices is at least (e?o(1)) (n!/2^n?1). © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 291–296, 2001     

Towards Washout Filter Concepts for Motion Simulators on the Base of a Stewart Platform

In this paper we present a pneumatically driven Stewart platform as a basis for motion simulators. Motion platforms, that simulate perceived situations in aeroplanes, cars or ships, have workspace constraints in every degree of freedom. Therefore it is necessary to adapt the accelerations and angular rates in order to stay within the physical restrictions. For realizing a flight or ride simulator on the basis of the Stewart platform, “ Washout Filters” are used to change the signals of the open source software FlightGearand to minimize the sensation error between simulator and aircraft. Different filter concepts are implemented and evaluated. The platform presented in this document is a parallel kinematic robot, driven by fluidic muscles. The pneumatically actuated muscles are only able to produce tensile forces. Therefore a spiral spring from a passenger car is used to apply the compressive forces and torques. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Expressivism about Knowledge and the Value of Knowledge

The aim of the paper is to state a version of epistemic expressivism regarding knowledge, and to suggest how this expressivism about knowledge explains the value of knowledge. The paper considers how an account of the value of knowledge based on expressivism about knowledge responds to the Meno Problem, the Swamping Problem, and a variety of other questions that pertains to the value of knowledge, and the role of knowledge in our cognitive ecology.     

Independent sets in regular graphs and sum-free subsets of finite groups

It is shown that there exists a function∈(k) which tends to 0 ask tends to infinity, such that anyk-regular graph onn vertices contains at most 2^{(1/2+∈(k))n} independent sets. This settles a conjecture of A. Granville and has several applications in Combinatorial Group Theory. Research supported in part by the United States-Israel Binational Science Foundation and by a Bergmann Memorial Grant.