What must have been allowed by the romans for christianity to spread so far and wide?

By the third century, Christianity was well established in and around Greece and the Middle East, as well as in Rome, Alexandria, Carthage and a few cities such as Lyons in the 'barbarian' western Europe.

Christianity had largely failed to penetrate Egypt outside Alexandria, or much of western Europe. Even Italy, outside the city of Rome, seems to have largely resisted Christianity. It seems that the Egyptian and Celtic religions had not entered a period of decline and scepticism in the way that the Greco-Roman religion had done. However, there was no impediment to Christians preaching in those areas, other than a lack of interest on the part of the population.

Christian tradition suggests that the Christians suffered constant harrassment and persecution by the Roman authorities. However, Euan Cameron (Interpreting Christian History: The Challenge of the Churches' Past) says, "Contrary to popular tradition, the first three centuries of Christianity were not times of steady or consistent persecution. Persecution was sporadic, intermittent, and mostly local." Edward Gibbon (The Decline and Fall of the Roman Empire) goes further and, on a number of occasions, praises the pagan Romans for their general tolerance towards Christianity. Widespread and persistent persecution of other faiths only really began with the Christian Empire.

There was a total of perhaps 12 years of official persecution of Christianity during nearly three hundred years in which Christianity existed in the pagan Empire. Otherwise, the Christians were largely allowed to worship as they pleased, and even to proselytise their faith, as long as they took care not to offend others or disturb the peace. This allowed Christianity to prosper and spread far and wide.

Hope this helps :)