Introduction:
The size of Built-Up Edge (BUE) in metal cutting refers to the accumulation of workpiece material on the cutting tool during the machining process. BUE formation can have a significant impact on the machining process and the quality of the machined surface. It is important to understand the factors that influence the size of BUE in order to optimize the metal cutting process.

Factors affecting the size of BUE:

1. Very high speed:
- High cutting speeds can lead to an increase in the size of BUE.
- At high speeds, the cutting tool encounters more resistance from the workpiece material, leading to higher temperatures at the cutting edge.
- The high temperatures soften the workpiece material, causing it to adhere to the cutting tool and form a larger BUE.

2. Large uncut chip thickness:
- The uncut chip thickness refers to the thickness of the material being removed by the cutting tool.
- When the uncut chip thickness is large, there is a higher chance of material build-up on the cutting tool.
- The large uncut chip thickness provides a greater contact area between the workpiece material and the tool, facilitating the formation of a larger BUE.

3. Use of cutting fluid:
- Cutting fluids are commonly used in metal cutting operations to improve tool life, reduce friction, and remove heat from the cutting zone.
- However, the use of cutting fluid can also have an impact on the size of BUE.
- In some cases, cutting fluids can reduce the size of BUE by lubricating the cutting zone and preventing the workpiece material from adhering to the tool.
- However, in certain situations, cutting fluids can promote the formation of BUE by facilitating the transfer of workpiece material onto the cutting tool.

4. Increase in positive rake angle:
- The rake angle of a cutting tool refers to the angle between the cutting edge and the surface being cut.
- A positive rake angle means that the cutting edge is inclined in the direction of the cutting force.
- An increase in the positive rake angle can lead to a larger BUE.
- This is because a higher positive rake angle increases the contact area between the workpiece material and the cutting tool, promoting the formation of BUE.

Conclusion:
Among the given options, the size of BUE in metal cutting increases with a large uncut chip thickness. This is because a larger uncut chip thickness provides a greater contact area between the workpiece material and the cutting tool, facilitating the formation of a larger BUE. The other options, such as very high speed, use of cutting fluid, and increase in positive rake angle, may have their own impacts on the metal cutting process, but they do not directly influence the size of BUE as significantly as the uncut chip thickness does.