Let ${{G}_{1}},\,{{G}_{2}},\,.\,.\,.\,,\,{{G}_{t}}$ be arbitrary graphs. The Ramsey number $R\left( {{G}_{1}},\,{{G}_{2}},\,.\,.\,.,{{G}_{t}} \right)$ is the smallest positive integer $n$ such that if the edges of the complete graph ${{K}_{n}}$ are partitioned into $t$ disjoint color classes giving $t$ graphs ${{H}_{1}},\,{{H}_{2}},\,.\,.\,.\,,\,{{H}_{t}}$ , then at least one ${{H}_{i}}$ has a subgraph isomorphic to ${{G}_{i}}$ . In this paper, we provide the exact value of the $R({{T}_{n}},\,{{W}_{m}})$ for odd $m,\,n\,\ge \,m-1$ , where ${{T}_{n}}$ is either a caterpillar, a tree with diameter at most four, or a tree with a vertex adjacent to at least $\left\lceil \frac{n}{2} \right\rceil \,-\,2$ leaves, and ${{W}_{n}}$ is the wheel on $n\,+\,1$ vertices. Also, we determine $R\left( {{C}_{n}},\,{{W}_{m}} \right)$ for even integers $n$ and $m,\,n\,\ge \,m\,+\,500$ , which improves a result of Shi and confirms a conjecture of Surahmat et al. In addition, the multicolor Ramsey number of trees versus an odd wheel is discussed in this paper.

The study of the phenomenon of cavitation of centrifugal pumps remains of a greatimportance in the field of construction as well as in the field of installation of pumpsin pumping stations. In fact, with the consequences it engenders: noise, vibration,erosion and loss of efficiency, it provokes, on the one hand, a progressive degradation ofthe performance of the machine and, on the other hand, an increase in the costs of themaintenance of installation (a compromise between the functioning period and cost ofexploitation), in addition to the loss of efficiency of pumps and the resistance ofdifferent materials to the effects of the phenomenon of cavitation. Among the techniquesand methods of protecting the parts of pumps is the cladding of the parts with compositeplastic materials for the purpose of increasing the resistibility and protecting the partsfrom the effects of cavitation. Relevant to this, our work aims at analysing this problemwith the intention of preserving the efficiency of a pump by limiting the effects ofcavitation through an experimental study that has been realized in a laboratory. As such,wheels of different materials are tested under conditions of cavitation taking intoconsideration the three types of metals used in the manufacture of the wheel, and thisregarding the study of resistibility of each material subject to the effects of thephenomenon aforementioned. Thus, this present work consists in experimentally studying thephysical impact of cavitational erosion upon the hydrodynamic behavior and theperformances of centrifugal pumps, and that affects the weight of different wheels. Thisphenomenon can be obtained only through measurement: loss of efficiency that can haverepercussions for the characteristics of functioning as well as the mechanical behavior.

This paper provides a strengthening of the theorems of small cancellation theory. It is proven that disc diagrams contain 'fans' of consecutive 2-cells along their boundaries. The size of these fans is linked to the strength of the small cancellation conditions satisfied by the diagram. A classification result is proven for disc diagrams satisfying small cancellation conditions. Any disc diagram either contains three fans along its boundary, or it is a ladder, or it is a wheel. Similar statements are proven for annular diagrams.

2000 Mathematical Subject Classification: 20F06.