Here is a very good proof that simultaneity is absolute, not relative. The question: Is this proof wrong, or is Einstein? Can you prove it? What is the mistake in the proof, if any? If there is none, then simultaneity must be absolute:
(a) Production of absolutely simultaneous events: Take the following apparatus: a battery and switch placed at the mid-point X of two event locations A and B. Connect the switch with two long pairs of wires of equal length from X to A and X to B, and then to a lamp at A and B which produces a flash of light. The lengths of wires, lamps and their connections are assumed to be of exactly identical construction, so that when activated by a single switch, both lamps produce the flash of light after exactly the same time delay, i.e. both flashes are produced simultaneously. So far, we have assumed only one frame of reference. But now mount this apparatus on a moving train (assume the train to be long enough to support the wires and lamps from A to B) and activate the switch, and the flashes at A and B will still be simultaneous. They will also be determined (see Einstein's method below) as simultaneous from ground - they have to be because as seen from the ground, both lamps are activated by the same switch, and as measured by a clock on the ground, both lamps produce the flash after the same delay for A and B, since the construction of both is identical. Agreed that the clock on the train may be running slowly compared to the clock on the ground, so the time delay between pressing the switch and production of the flash as seen from the ground may be different, but it will be the same for A vs B and so the events will still be simultaneous. Also when mounted on the train, the events will be produced at different locations A' and B' instead of A and B, but still, the events at A' and B' will be simultaneous when seen from the ground (and from the train). Therefore, it is possible to generate events which are simultaneous in all inertial frames of reference.
(b) Determination of absolutely simultaneous events: We use Einstein's method from Einstein's book, chapter "Relativity of Simultaneity". If an observer at the mid-point X of event locations A and B observes that the events are simultaneous, then we conclude that the events are simultaneous. I feel quite sure that Einstein will agree that it is important that the observer must be at the mid-point X at the instant when he observes both events. If the observer is at some other location yesterday, or tomorrow, or even if he is at some other location at the same instant as when the flashes of light are actually produced at A and B, then all that is irrelevant to our experiment. We know that at the instant the flashes are produced at A and B, the observer may be somewhere else, because the light from the flashes has not reached him yet. But at the instant when light from both flashes reaches the mid-point X, and the observer is there to observe it, he will determine both flashes as simultaneous, because light from both will reach the mid-point simultaneously. We note that even in a single frame of reference, only the observer at the correct location can determine the events to be simultaneous. Observers at A or at B or at any other location cannot make that determination.
Now consider the simultaneous flashes on the ground and Einstein's observer on the train. The mistake Einstein made in his proof was the he assumed the observer to be at the mid-point X at the instant the flashes occurred. (assuming the train is going in direction from B to A) So of course, the light from A meets the observer first because the observer has travelled closer to A by the time the light meets him, and the light from B meets the observer later because he has moved away from B. He is no longer at the mid-point when he makes the observations. This can only be described as a flawed experimental technique which makes the actually simultaneous events look like non-simultaneous.
Instead, if we station a large number of observers on the train and the train is long enough, there will be one observer on the train, who will view both flashes as simultaneous, and that observer will be at the mid-point X at the instant when he makes his observation -obviously even though he is travelling on the train, if he is at the mid-point X at the instant when light from both A and B reaches him, both will reach him simultaneously and he will view both events as simultaneous. So for the observer who is in a position to make a correct observation, the events are simultaneous for observers, both on the ground and on the train. In fact, the events are simultaneous for ALL inertial frames in uniform motion relative to each other, provided their observers are at the correct location to view the events.
Observers who are not at the mid-point between A and B at the instant of the observation would observe the events as non-simultaneous not because the events are non-simultaneous in their (moving) frame of reference but simply because light takes unequal time to travel the unequal distance from A and B to the observer. That obviously says nothing about whether the events are really simultaneous or not. If this effect was to be called "Relativity of Simultaneity", then simultaneity would be relative even in the same frame of reference as the events. Observers in the same frame of reference as the events, who are closer to A will view A before B and observers closer to B would view B first. But that does not mean that A and B are not simultaneous. As given above, they can be generated simultaneously, so they are provably simultaneous. The different observations by different observers can only be attributed to observation error due to light from A and B travelling different distances to the observer in different amounts of time. The same observation error is also there for an observer in another frame moving with respect to the frame of the events. Why should we then say that the events are non-simultaneous in another frame, or that simultaneity is relative?