On-line Ramsey numbers

Consider the following game between two players, Builder and Painter. Builder draws edges one at a time and Painter colours them, in either red or blue, as each appears. Builder's aim is to force Painter to draw a monochromatic copy of a fixed graph G. The minimum number of edges which Builder must draw, regardless of Painter's strategy, in order to guarantee that this happens is known as the on-line Ramsey number \tilde{r}(G) of G. Our main result, relating to the conjecture that \tilde{r}(K_t) = o(\binom{r(t)}{2}), is that there exists a constant c > 1 such that \tilde{r}(K_t) \leq c^{-t} \binom{r(t)}{2} for infinitely many values of t. We also prove a more specific upper bound for this number, showing that there exists a constant c such that \tilde{r}(K_t) \leq t^{-c \frac{\log t}{\log \log t}} 4^t. Finally, we prove a new upper bound for the on-line Ramsey number of the complete bipartite graph K_{t,t}.Comment: 11 page

Combinatorial theorems relative to a random set

We describe recent advances in the study of random analogues of combinatorial theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201

Embedding Product Life-cycle Management (PLM) in Higher Education: A Case Study in Fashion Business

This study represents one of the first attempts to embed a PLM philosophy and system within an undergraduate course. What is sought through this intervention is the development of a critical mindset capable of utilising technological innovations to critique current / traditional processes and practices and create alternatives that respond to the demands and opportunities of new times, new needs and changes in circumstances