The famous Aihole inscription of Chalukya King Pulkeshi II says: the temple was constructed 3735 years, after the Bharat War. Thus the date of the MB war by this inscription is 3102 BCE.This inscription was written too long after the MB war, and hence this is not given much credence.In 500 CE, Aryabhatta calculated the start of Kali Yug as 17th February, 3102 BCE.
This is also proved by the fact that, the kings like Sudas, Divodasa, Somaka, etc. As the Puranic genealogies are far from complete, the date of 1924 BCE is the lower limit for the MB date.(I am writing "should" because it is not yet discarded from the history text books.) Astronomical events mentioned in MB are used to calculate the date of the MB War.One such event is the death of Bheeshma on a winter solstice day, Magh Shukla Ashtami.From this date one calculates the date of MB as follows: The axis of the earth is tilted to the ecliptic (the plane in which the Earth revolves around the sun) by 23.5 deg.For an observer on the Earth, it appears as if the Sun is in the North in summer, and in the South in winter.