Due to Brouwer fixed point theorem in any rotating disk, the center remains motionless (i.e.fixed point) , therefore observer on this point actually meets an inertial reference frame, However any other reference frames on other points bearing a non inertial reference frame because of their circular motion respect to center inertial reference frame. On the other hand you should not  forget the fact that, you, as the narrator of this setting play a role as the third observer, because the rotating disk has been verified in your own reference frame where the center may be chosen at rest.   

For a disc in an otherwise empty universe, the concept of “rotation” of the disc (or any kind of “motion” of the disc is undefinable. Movement of an object is defined relative to other objects. This led Mach to his “principle” – that the set of inertial frames is determined by the distribution of all the matter in the space. This is intuitively appealing (and probably correct), but has never been given a satisfactory theoretical foundation. Einstein was hoping that his General Theory of Relativity would “explain” Mach’s principle. Whether it does or not is debateable.

Happy new year and thank you all for your answers!

@Manouchehr Amiri There is no need of a third observer for reference. The essential question is the following: Does observer B really measure an actual inertial force because of his rotation? Observer A meets an inertial reference frame as you said and he knows that because he can’t measure any external force. 

@Eric Lord I can understand that due to Mach’s principle there is no centrifugal force on B and therefore there is no absolute but a relative rotation (obs. A think that B is rotating and obs. B think that A is rotating). But I really can’t understand how rotation is undefinable. We could define it mathematically.

If Mach’s principle is wrong, does we have to admit that there is absolute rotation?Because this may be another good reason to accept it’s validity.

@Theophanes E. Raptis I disagree with this rocket example, because rockets need mass (fuel) to jet away in order to rotate the disk.

Theophanes ~

What a puzzling question. If Newton’s second law is taken as valid we get the strange conclusion that it has an angular acceleration, but that, because there is no reference point outside the system, its angular velocity is undefinable! On the other hand, in order to apply Newton’s second law to this situation, we have to assign a mass (and consequently a moment of inertia) to the disc. The question is then whether it makes sense to speak of the mass of the disc if there is nothing else in the universe. If Mach is right in his guess that inertial phenomena (including the inertial mass of the disc) are a consequence of all the matter in the universe, we cannot. Unfortunately, we have only one universe, which is not a universe that’s empty apart from a disc, so there’s no possibility of experimenting to test our speculations!

@Theophanes E. Raptis. You are right in “additional premise of massless”. After all, if an object is fixed on the edge of disc, the acceleration of this object is set by the disc rotation, and the centrifugal force is proportional to the mass of the object. Accordingly, a small sinker on the end of the thread (tying it to point B) will pull this thread on the radius from point B only if the sinker has a mass!

Therefore, in the initial question all objects must have a small mass.

@Tilemachos M. Athanasiadis . There is no Mach’s principle in centrifugal force. A small sinker on the end of the thread (tying it to point B) will pull this thread on the radius from point B only if the sinker has a mass, but here there is no need to have any masses around the disc!

Dear Tilemachos, Happy new year, In response to your comment, I recall the especial relativity axiom that states: all inertial reference frame are equivalent in formulation of physical laws, or physical laws formalism are covariant (i.e. the same) respect to inertial frames.When you speak about a rotating disk it should be verified in a reference frame,  just like the any other motions, you need a reference to define a motion, However if you insist on the uniqueness of a disk , and  you are limited to the references on this disk, then if an observer on a point separate from center, requires a centripetal force to maintain its location on the disk ( due to your story this disk is the only reference frame) then the all observers on such points conclude that the disk is rotating and unique inertial point (reference) will be the center of rotation. this is consistent with Relativity principle (or axiom) that the observers on an inertial frame can not be aware of its relative motion respect to other references (when they limited to their own references. However in a non-inertial reference (such as your disk) some experiment of observers lead to finding out the rotation of the disk (or their reference frame). i remind again that in your disk there are two type of reference frames 1) center of disk as inertial frame and  2) other points as non-inertial points.

It may seems that the problem has to do with relativity in the sense that with  no reference to the exterior of the disc  it  may not feel the centrifugal force. Our daily life takes place on the edge of a rotating disc (land area) that provides us with a centrifugal force (due to the rotation of the earth) which we do not perceive in any way without reference to the outside of the disc (stars, sun, planets). General Relativity is a local theory.

Please read the paper, may be helpful for you.  A new approach to the correction of

Galilean transformation.

Dear Tilemachos M. Athanasiadis

Simple in my view. If you consider absolute motion, then the amount of momentum energy of each observer will be the reference.

The observer at position A will have much less momentum energy than the observer at position B. Each observer verifying and confirming their respective amounts of momentum energy and comparing them, will be able to determine which observer is at the center of rotation and which one is at the periphery of the disk.