Alexander the Great's empire extended to which river in the East?

The Indus, the longest river of Pakistan, more than 3,000 km. It flows from Tibet to the Arabian Sea near Karachi.

An empire is an ill-defined institution. It generally implies that one anointed leader rules over a number of different nations united by a common allegiance. Whether Alexander actually built an empire is a point debated among historians. He certainly won all his battles, founded a great number of cities and conquered vast territories. But did he actually rule? Did the conquered nations renounce sovereignty and swear fidelity to him?

Alexander did not even make it back home. He died ten years after the beginning of his Asian military campaign while on his way back to Macedonia. He did not have the time (the chance?) to organise, normalise and unify with appropriate legislation and planning the vast territory he had subjugated. He left behind 23 cities named after himself but not a Capital. He appointed a number of rulers (mostly his generals) but they governed independently from one another and often in open conflict among themselves. Soon after Alexander death, each of them became an oriental satrap at the origin of oriental dynasties: the Seleucids, the Ptolemais, the Antipater, the Soliderchus…

In fact Alexander's heritage never was an empire; but a most powerful culture, that we call Hellenistic, that peacefully, yet inexorably, followed him wherever he brought his sword.

The Greeks had invented a way of living, of thinking, of representing themselves and of shaping their habitat unknown to other nations. Within what we call the Western World, they had invented philosophy, music, poetry, geometry, astronomy… all disciplines that had little to do with military might and all to do with intelligence, art and beauty. This was the message that Alexander's army brought as far as India and Pakistan. While his ephemeral conquests did not last the turn of a single century, the remnants of the Hellenic culture that constituted the backbone of each one of his soldiers, are continuously dug-up by stupefied archeologists in the most improbable and remote regions of his fleeting empire.