$1.4.3.$ One of the dust cloud particles (particle $A$) is at rest, and all the others are flying away from it in different directions with velocities proportional to the distances from them to particle $A$. What motion pattern will an observer detect when moving with particle $B$?
He will find that all particles are moving away from him with a speed directly proportional to the distance from particle $B$ to them.
This can be proven as follows: write down the speeds of particle $B$ and an arbitrary particle $C$ in the reference frame of particle $A$, and then go to the reference frame of particle $B$, and make sure that the speed of particle $C$ in this reference frame is proportional to the distance between particles $B$ and $C$. Moreover, it is proportional with the same coefficient as in the reference frame of particle $A$.
Exactly the same as an observer moving with particle $A$.