Demetrius' victory over Ptolemy at Salamis in 306 gave him and his father Antigonus Monophthalmus so much credit, that it was widely believed that they would reunite the empire of Alexander the Great. With some justification, the two men accepted the royal title; after all, the descendants of Alexander were by now all dead. The other Diadochi followed immediately.
The story is told by Plutarch of Chaeronea in his Life of Demetrius 18. The translation was made by M.M. Austin.
The Diadochi assume the royal title
[18.1] The multitude then for the first time proclaimed Antigonus and Demetrius kings.note[Antigonus was in Syria. The demands of the crowd were carefully prepared. Demetrius was at Salamis on Cyprus.] Antigonus' friends tied at once a diadem round his head, while Demetrius was sent a diadem by his father and addressed as king in a letter he wrote. When the news was reported, Ptolemy's followers in Egypt also proclaimed Ptolemy king, to dispel any impression that his defeat had humbled his pride.
[18.2] And so emulation spread the practice like a contagion among the Diadochi: Lysimachus began to wear the diadem, and so too Seleucus in his dealings with the Greeks (with the barbarians he had already been behaving as a king). Cassander, however, although the others wrote to him and addressed him as king, continued to write letters in the same style as before, with his name only but no title.
[18.3] Now this practice did not involve merely the addition of a title and a change of fashion; it stimulated the men's pride and raised their ambitions, and made them arrogant and obnoxious in their style of living and in their dealings with others. It is the same as with tragic actors who change their step, their voice, their posture at table and their way of addressing others when they put on their costumes.
[18.4] They became harsher in their judicial verdicts and no longer concealed their power, which had often in the past made them more lenient and gentle with their subjects. Such was the power of a single word spoken by a flatterer, and so great was the revolution it brought about in the world.