The size of a gas particle is negligibly small. The average kinetic energy of a particle is proportional to its temperature in kelvins. The collision of one particle with another (or with the walls of its container) is completely elastic. Which postulate(s) are needed to explain the increase in pressure of a gas with temperature?

The gas is composed of small molecules are they are in continuous random motion and having elastic collisions with one another and also with the walls of the container.
The molecules of the gas does not exert any kind of repulsive or attractive forces on each other and they their size is negligible as compared to the difference between them.
Pressure exerted by the molecules of the gas results from the collisions which is happening between the molecules of the gas and the walls of the container.
Average kinetic energy of molecules of the gas is directly proportional to absolute temperature.

Out of the given postulates, the postulate which can be used to explain the increase in pressure of a gas with temperature is:- The average kinetic energy of a particle is proportional to its temperature in kelvins.

This is because increase in the kinetic energy demonstrate that the molecules are moving faster which leads to greater pressure. Thus, Pressure is directly proportional to the kinetic energy which in turn is directly proportional to temperature.

So, Pressure is directly proportional to the temperature.