Given a group of people, some of whom are connected in a particular way, a natural question to ask is: how many links does it take to connect any person to any other, or to a specified 'base' person? Mathematics has the "Erdos number": someone has an Erdos number of 1 if they co-wrote a research paper with Paul Erdos, 2 if they didn't write one with Erdos but did write one with someone else who wrote one with him, and so on.
Cinema has the similarly defined "Bacon number": someone with a Bacon number of 1 appeared in a film alongside Kevin Bacon, someone with a Bacon number of 2 appeared alongside someone who appeared alongside Bacon. What if we consider a similar concept for cricket? Erdos and Bacon were chosen as the 'base' individuals for their respective disciplines because they had a large number of connections: Erdos had more co-authors than any other mathematician in history, and Bacon a similarly high number of co-actors - so it makes sense for the 'base' individual in cricket to be the one who has played either with or against more players than anyone else in the history of international matches; unsurprisingly, this is Sachin Tendulkar.
So we define:
T(Tendulkar) = 0 (the 'base' person in any similar system is the only one to have a number of 0) and
T(X) = n if X has played in an organised cricket match either with or against a player Y such that T(Y) = n-1, and n is the lowest number for which this is the case.
The wording "any organised cricket match" in the definition allows players who've never appeared in an international or even first-class match to claim Tendulkar numbers, whilst disqualifying the most tenuous links - bowling a few balls at someone in the nets does not count as having played against them. By the end of India's home series against Australia, Tendulkar had played either with or against 982 others in international matches, with Ajinkya Rahane the latest addition to the list after making his Test debut in Delhi (Rahane had previously played 16 ODIs and seven T20Is, but none of them alongside Tendulkar); add in domestic FC, T20 and a few guest appearances in exhibition matches below first-class level, and there are most likely around 1500 players in total with a Tendulkar number of 1.
The oldest of these is probably John Traicos (born 1947), the Egyptian-born, Greek-descended offspinner who played three Tests for South Africa before Tendulkar was born, and was still good enough at the age of 45 to be selected for Zimbabwe's inaugural Test against a 19-year-old Tendulkar who already had three Test centuries to his name; experience triumphed over youth, with Traicos taking five wickets in India's only innings including Tendulkar caught and bowled for a duck.
At the other end of the age range, Kragg Brathwaite wasn't born when Tendulkar and Traicos faced off, but faced Tendulkar himself when West Indies toured India in 2011.
The players with a Tendulkar number of 2 are those who haven't played the man himself, but have played someone else who has; they include everyone who's appeared in a Test since his debut without playing him (but not everyone who's appeared in an ODI in that time: a few Scottish players have made their only ODI appearances against Canada and vice versa, and since Tendulkar has never played either of those teams, they haven't played anyone who's played him), as well as countless players at lower levels thanks to Test players making guest appearances for club teams.
Almost certainly the oldest player with a Tendulkar number of 2, and one of the most unexpected, is Keith Miller (born 1919), who made his Test debut before Traicos was born but, 20 years after playing his last Test, appeared in a series of exhibition matches for an International XI against a Pakistan XI in November 1976, the last of them on his 57th birthday; the Pakistan XI included Javed Miandad, who later played Tests against Tendulkar. Miller's comeback wasn't a success; he scored one run in two innings and didn't bowl.
Even the lowest level current club players are likely to be able to claim a Tendulkar number of 3 or 4: perhaps one of the best players in your local league played at a higher level alongside a first-class player, who played against a Test player who played Tendulkar, or something along similar lines. Thanks to a few players with remarkably long careers - and those, such as Miller, who played exhibition matches long after their retirement from the top level - many players from previous eras can claim a lower number than might be expected.
CK Nayudu (born 1895) may be the only player with a Tendulkar number of 3 who was born in the 19th century; on his final first-class appearance in 1963 at the age of 68 he faced Ajit Wadekar, who never played a Test against anyone who played Tendulkar, but did play against Miandad in the same series as Miller in 1976. Nayudu's first-class debut, 47 years earlier, came a few months after the death of WG Grace, but the opposition included Frank Tarrant, who had played against Grace, giving Tarrant a Tendulkar number of 4 and WG one of 5.
Tarrant had a pretty long career himself - starting in the closing years of the 19th century and finishing only in the final of the Bombay Quadrangular Tournament in 1936. Among the opposition in that match was Vijay Merchant, who gives another chain linking Tarrant to Tendulkar in four steps: Merchant played his last Test in 1951, against an England team including Derek Shackleton, whose county career lasted until 1970, when he turned out for Hampshire with Gordon Greenidge; Greenidge and Tendulkar never met in a first-class match, but did in an exhibition match between India and a Rest of the World XI in 1994.
Don Bradman played alongside Miller, giving him a Tendulkar number of 3, and was part of the Australian team which toured the US and Canada in 1932. In the course of that tour Bradman appeared against a team of "British Born Film Stars", captained by the then 69 year old Aubrey Smith - who captained England in what would later be recognised as South Africa's first (and Smith's only) Test, before going on to a rather lengthier career in Hollywood. Smith thus has a Tendulkar number of 4 - and was a regular in county cricket in the 1880s and 1890s, giving an alternative link to WG as well as Tendulkar numbers of 5 for many other players of that era.
Thanks to Grace and Jack Blackham - who played in every one of the first 17 Tests - every player who's ever appeared in an international match has a Tendulkar number of at most 7. How high can the numbers go and how far back in cricket history can players be linked? The answers turn out to be fairly high, and a very long way back.
The earliest matches from which the names of the players survive are London vs Slindon and England vs Kent in 1744, but none of the players in those two matches is known to have played in any later year, and so cannot be linked to any later players. The next earliest, however (Surrey vs Hampshire in 1768), featured John Small, one of the leading players of the 18th century. He is linked to Tendulkar by a chain of 9, not including WG Grace but his brother EM: Small played with Billy Beldham, who played with John Sherman, who played with George Chatterton, who played with EM Grace, who played with Aubrey Smith, who played with Bradman, who played with Miller, who played with Miandad, who played with Tendulkar. The other players in the 1768 match, who played with Small but not with Beldham, have Tendulkar numbers of 10.
John de Burgh, later the 13th Earl of Clanricarde, played a solitary first-class match in 1773 in which Small was not involved (but others who had played with him were).; thus de Burgh, along with any other players of that era who never played with Small, has a Tendulkar number of 11 - which is almost certainly the highest such number of anyone.